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+{"text":"\\section{Introduction}\n\nThe main statement of the Runge--Gross theorem is that the potential to density mapping is invertible modulo a time-dependent constant in the potential~\\cite{RungeGross1984}. This statement implies that knowledge of the time-dependent density and the initial state is in principle sufficient to fully characterize the evolution of a quantum system. The invertibility theorem by Runge--Gross therefore forms the cornerstone of time-dependent density functional theory (TDDFT)~\\cite{tddft2006, CasidaHuix-Rotllant2012, Ullrich2012, tddft2012}. Unfortunately, the Runge--Gross theorem only holds for potentials which are Taylor-expandable in time. \nTaylor expandability of the potential is actually a too stringent condition and can be loosened as demonstrated by the invertibility theorem for linear response by Van Leeuwen~\\cite{Leeuwen2001} and more recently, by work of Tokatly on lattice systems~\\cite{Tokatly2011a, Tokatly2011b, FarzanehpourTokatly2012} and the fixed-point approach by Ruggenthaler and Van Leeuwen~\\cite{RuggenthalerLeeuwen2011, RuggenthalerGiesbertzPenz2012, RuggenthalerPenzLeeuwen2015}.\n\nPractical TDDFT calculations are almost exclusively performed with the help of an auxiliary non-interacting reference system, the Kohn--Sham system~\\cite{RungeGross1984, KohnSham1965}, which has the same density as the fully interacting system thanks to the exchange-correlation potential. The exact exchange-correlation potential depends on all densities at earlier times and on the initial states in a complicated manner~\\cite{RuggenthalerNielsenLeeuwen2013}. This history dependence of the exact exchange-correlation potential is neglected in practice and replaced by its ground state version in which the instantaneous density is inserted. This neglect of memory dependence is known as the adiabatic approximation and the results are typically very satisfactory for polarizabilities and local valence excitations. Especially when a good model for the exchange-correlation potential is used even Rydberg excitations can be reproduced reliably~\\cite{GisbergenKootstraSchipper1998, SchipperGritsenkoGisbergen2000, AppelGrossBurke2003}. Practical TDDFT calculations fail for more complicated excitations such as charge transfer excitations~\\cite{DreuwWeismanHead-Gordon2003, GritsenkoBaerends2004b} and bound excitons~\\cite{RohlfingLouie1998, BenedictShirleyBohn1998}, when the hole and electron are not localized close to each other~\\cite{BaerendsGritsenkoMeer2013}, though some progress has been reported in their TDDFT description~\\cite{ReiningOlevanoRubio2002, YanaiTewHandy2004, YangUllrich2013}. Even more problematic are double~\\cite{MaitraZhangCave2004, CaveZhangMaitra2004} and bond-breaking~\\cite{GritsenkoGisbergenGorling2000, GiesbertzBaerends2008} excitations. The main problem is that the density is not a natural quantity to describe these excitation processes and the non-interacting Kohn--Sham system is also of no avail.\n\nA more natural quantity to deal with these more complicated physical processes is the one-body reduced density matrix (1RDM). In particular its fractional occupation numbers are good descriptors of correlation effects. Indeed, it has been demonstrated that time-dependent 1RDM functional theory is capable of correctly describing charge-transfer excitations, double excitations and bond-breaking excitations~\\cite{GiesbertzBaerendsGritsenko2008, GiesbertzPernalGritsenko2009, GiesbertzGritsenkoBaerends2010a} even within the adiabatic approximation.\nUnfortunately, no proper formal justification for time-dependent 1RDM functional theory has yet been published. The main purpose of this paper is to partially eliminate this caveat by presenting an invertibility theorem for non-local potentials and the 1RDM in the linear response regime.\n\nIn previous work~\\cite{PernalGritsenkoBaerends2007} one tried to avoid the lack of a formal foundation by invoking the the Runge--Gross theorem. If all observables are functionals of the time-dependent density, they certainly are of the 1RDM, since the density can readily be extracted from the 1RDM. The problem is that there are (infinitely) many 1RDMs generating the same density, but only one of these 1RDMs corresponds to the local potential belonging to that density, the `local' 1RDM. So if we want to use the Runge--Gross theorem (or one of the extensions) as a foundation for time-dependent 1RDM functional theory, we are only allowed to use these `local' 1RDMs. Since the characterization of these `local' 1RDMs is seems to be impossible, direct use of the Runge--Gross theorem does not lead to any viable theory.\n\nActually, if we follow the philosophy of TDDFT more closely, we should not consider the mapping from local potentials to 1RDMs, but from non-local potentials to 1RDMs, since the non-local potential is the natural conjugate variable of the 1RDM. With a non-local potential I mean a one-body potential non-local in space, but still local in time. A generalization of the Runge--Gross theorem from local potentials and densities to non-local potentials and 1RDMs would therefore be more appropriate. Unfortunately, a straightforward generalization is not possible, since the commutator between the 1RDM and the interaction does not vanish, $\\bigl[\\hat{\\gamma},\\hat{W}\\bigr] \\neq 0$.\n\nThe invertibility theorem for the density response function for Laplace transformable potentials by Van Leeuwen~\\cite{Leeuwen2001} is much more amenable to generalization to the 1RDM. Assuming that the initial state is a non-generate ground state and that the time-dependent part of the potential is Laplace transformable, the theorem by Van Leeuwen states that the density response function is invertible up to a constant shift in the potential. The proof by Van Leeuwen can be split into two parts. The first part consists of the derivation of a necessary and sufficient condition for a perturbation to yield zero response, i.e.\\ necessary and sufficient condition for a potential to belong to the kernel of the response function. This first part does not need the special properties of the density operator and can therefore readily be generalized to arbitrary operators, e.g.\\ the 1RDM operator, the current operator, the (non-collinear) spin density operator and the kinetic energy density operator. The second step of the proof by Van Leeuwen is to check the necessary and sufficient condition for the density operator, so the second part depends on the particular properties of the density. A generalization of the second step to arbitrary operators is therefore not possible, but needs to be considered separately for each operator. In this article I will work out the second step for the 1RDM operator, which allows me to completely characterize the kernel of the 1RDM response function.\n\nThe first part of the proof by Van Leeuwen requires at one point that the initial ground state is non-degenerate. This non-degeneracy requirement is quite a nuisance, since this requirement immediately excludes all open shell systems. I will not only repeat the first part of the proof by Van Leeuwen for arbitrary operators, but also show how the theorem can be extended to include degenerate initial states. Including initial degenerate ground states leads to an additional condition which needs to be checked. I work out this additional condition for the density operator and show that the inclusion of degenerate states does not lead to additional potentials in the kernel of the density response function. This provides an extension of the invertibility theorem for the density response function by Van Leeuwen to degenerate ground states. Degenerate ground states are also considered for the 1RDM response function.\nIncluding the possibility of degeneracy leads to a more general expression for some of the potentials in the kernel of the 1RDM response function found non-degenerate case.\nThese results put linear response time-dependent 1RDM functional theory on a rigorous foundation. A more complete theoretical framework would be achieved by also addressing the $v$-representability question. The generalized invertibility theorem only determines to which extent the perturbations $\\delta v$ yielding $\\delta\\gamma$ are unique. It does not answer the question if there actually exists a $\\delta v$ which yields the desired $\\delta\\gamma$, i.e.\\ the question of $v$-representability of $\\delta\\gamma$. For a complete theory, also the $v$-representability question should be addressed to specify exactly which $\\delta\\gamma$ we are allowed to use in linear response time-dependent 1RDM functional theory. This paper focusses on the invertibility of the response function, so the $v$-representability question is beyond the scope of this article.\n\nAs Van Leeuwen showed in Ref.~\\cite{Leeuwen2003}, his invertibility proof for the density response function also works in the time-independent case by taking the limit of the variable in the Laplace transform to zero. Especially for ground state 1RDM functional theory this is an important result, since a Hohenberg--Kohn like proof for the non-local potential to 1RDM mapping does not exist as pointed out by Gilbert~\\cite{Gilbert1975}. Using the invertibility proof for the 1RDM response function for the time-independent case I can give a full classification of the non-uniqueness of the non-local potential featured in 1RDM functional theory for the first time. Application to the time-independent case only works for non-degenerate ground states unfortunately, since degeneracies are treated in a fundamentally different manner in both cases.\n\nThe paper is organized as follows. I start by repeating the first part of the invertibility proof by Van Leeuwen for arbitrary operators and extend his approach to handle degenerate ground states as well. After necessary and sufficient conditions have been obtained to characterize the potentials which do not lead to a response, I work these conditions out for the density and the 1RDM operators. First I consider the density operator to check against the result by Van Leeuwen~\\cite{Leeuwen2003} and to extend his result for the density response function to degenerate ground states. Next, I consider the 1RDM operator to find the non-local potentials which do not lead to a response of the 1RDM. First I consider the simpler case of non-degenerate ground states and then take the additional necessary condition into account to extend the result to degenerate ground states. In the last part before concluding, I discuss the implications of these results for ground state 1RDM functional theory and point out in more detail why degeneracies need a different treatment in the ground state case.\n\n\n\n \n\n\n\n\\section{The generalized invertibility theorem}\nNow let us repeat the first part of the invertibility proof by Van Leeuwen~\\cite{Leeuwen2001} for arbitrary operators $\\hat{Q}_i$ and generalize the proof to degenerate ground states. The set of operators $\\{\\hat{Q}_i\\}$ can be any set of self-adjoint operators of interest, e.g.\\ dipole and quadrupole operators. The index $i$ is also allowed to be a continuous index to represent a self-adjoint operator density such as the density operator, $\\hat{n}(\\coord{r})$. Of course, a mixture of continuous and discrete indices is also allowed such as in the spin-density operator, $\\hat{n}(\\coord{x})$, or the 1RDM operator $\\hat{\\gamma}(\\coord{x},\\coord{x}')$, where $\\coord{x} \\coloneqq \\coord{r}\\sigma$ is a combined space and spin coordinate. I will use the symbol $\\coloneqq$ throughout the paper to emphasize definitions.\n\nHaving selected some set of self-adjoint operators and\\slash{}or operator densities of interest, we consider perturbations by these operator $\\hat{Q}_j$ with strengths $\\delta v_j(t')$. Note that we need $\\delta v_j(t') \\in \\mathbb{R}$ to ensure that the total perturbation remains hermitian. Now we consider the linear response of the expectation values of the same set of operators~\\cite{FetterWalecka1971}\n\\begin{align}\\label{eq:lineResp}\n\\delta Q_i(t) = \\sum_j\\binteg{t'}{0}{t} \\chi_{ij}(t-t')\\delta v_j(t') ,\n\\end{align}\nwhere $\\chi_{ij}(t-t')$ is the retarded\\slash{}causal linear response function which can be defined as\n\\begin{align}\\label{eq:respFuncDef}\n\\chi_{ij}(t-t') \\coloneqq -\\mathrm{i}\\theta(t - t') \\brakket{\\Psi_0}{[\\hat{Q}_{H_0,i}(t),\\hat{Q}_{H_0,j}(t')]}{\\Psi_0} .\n\\end{align}\nIn its definition we have used the operators in their Heisenberg representation with respect to the unperturbed Hamiltonian, $\\hat{Q}_{H_0,i}(t) \\coloneqq \\textrm{e}^{\\mathrm{i}\\hat{H}_0t}\\hat{Q}_i\\textrm{e}^{-\\mathrm{i}\\hat{H}_0t}$, which is often referred to as the interaction picture. We have also used the Heaviside function, which is defined as\n\\begin{align*}\n\\theta(x) \\coloneqq \\begin{cases}\n1 &\\text{for $x > 0$} \\\\\n0 &\\text{for $x < 0$}.\n\\end{cases}\n\\end{align*}\nThe retarded response function can alternatively be expressed as a sum-over-states (its Lehmann representation) as~\\cite{Lehmann1954, FetterWalecka1971, Leeuwen2003}\n\\begin{align}\\label{eq:sosRespFunc}\n\\chi_{ij}(t-t') \n&= \\mathrm{i}\\theta(t-t')\\sum_K\\textrm{e}^{\\mathrm{i}\\Omega_K(t-t')}{q_i^K}^*\\!q_j^K + \\text{c.c.} ,\n\\end{align}\nwhere $\\Omega_K \\coloneqq E_K - E_0 \\geq 0$ are excitation energies and $\\Omega_K = 0$ only for $K < D$, so $D$ denotes the multiplicity of the ground state degeneracy~\\footnote{The sum runs over a complete set of states, so also includes a possible continuum where the sum should be interpreted as an integral.}. Further, we have defined\n$q^K_i \\coloneqq \\brakket{\\Psi_0}{\\hat{Q}_i}{\\Psi_K}$.\nNote that the initial state can be excluded from the sum, since $q_i^0 = \\brakket{\\Psi_0}{\\hat{Q}_i}{\\Psi_0} \\in \\mathbb{R}$, because the operators $\\hat{Q}_i$ should be hermitian. Inserting the sum-over-state expression for the response function in~\\eqref{eq:lineResp}, the response of the expectation value of the operator $\\hat{Q}_i(t)$ can now be written as\n\\begin{align*}\n\\delta Q_i(t) = \\mathrm{i}\\sum_K{q^K_i}^*\\binteg{t'}{0}{t} a_K(t')\\textrm{e}^{\\mathrm{i}\\Omega_K(t-t')} + \\text{c.c.} ,\n\\end{align*}\nwhere we have defined\n\\begin{align}\\label{eq:aKdef}\na_K(t) \\coloneqq \\sum_jq^K_j\\delta v_j(t) .\n\\end{align}\nThe integral has the form of a convolution product over the interval $[0,t]$ and can be transformed into a normal product by taking the Laplace transform\n\\begin{align*}\n\\Laplace [\\delta Q_i](s) = \\mathrm{i}\\sum_K{q^K_i}^*\\frac{\\Laplace[a_K](s)}{s - \\mathrm{i}\\Omega_K} + \\text{c.c.} ,\n\\end{align*}\nwhere the Laplace transform is defined as\n\\begin{align*}\n\\Laplace [f](s) \\coloneqq \\binteg{t}{0}{\\infty} \\textrm{e}^{-st}f(t) .\n\\end{align*}\nNow we multiply this equation by the Laplace transform of the potential $\\Laplace[\\delta v_i](s)$ and sum over the index $i$ to obtain\n\\begin{align*}\n\\sum_i\\Laplace [\\delta v_i](s)\\Laplace [\\delta Q_i](s)\n= -2\\sum_K\\frac{\\Omega_K}{s^2 + \\Omega_K^2}\\abs{\\Laplace [a_K](s)}^2 .\n\\end{align*}\nIn absence of response, we have that $\\delta Q_i = 0$, so we also have that $\\Laplace [\\delta Q_i] = 0$ and we obtain from the previous equation that for zero response we necessarily have\n\\begin{align*}\n0 = \\sum_K\\frac{\\Omega_K}{s^2 + \\Omega_K^2}\\abs{\\Laplace [a_K](s)}^2 .\n\\end{align*}\nBecause $\\Omega_K \\geq 0$ and only for $K < D$ we have $\\Omega_K = 0$, all the contributions for $K \\geq D$ are positive. Therefore, one necessarily has $\\Laplace [a_K](s) = 0$ for $K \\geq D$, so $a_K(t) = 0$ for $K \\geq D$ as well. Strictly speaking, we should say that $a_K(t) = 0$ almost everywhere for $K \\geq D$, since $a_K(t) \\neq 0$ on a set of measure zero in time would not contribute to the integral of the Laplace transform.\nFrom its definition~\\eqref{eq:aKdef} it is clear that there are only two possibilities for $a_K(t) = 0$ almost everywhere. The first possibility is the absence of a perturbation, $\\delta v_j(t) = 0$ almost everywhere. If we further assume that we are only interested in the classical solutions of the Schr\u00f6dinger equation (these are the usual physical wave functions defined at each point in time), we have as an additional condition that the potential needs to be continuous up to its first order derivative in time, $\\delta v_j(t) \\in C^1$~\\footnote{In Ref.~\\cite{RuggenthalerPenzLeeuwen2015} it is stated that the condition $\\delta v_j(t) \\in C^1$ can probably be weakened to Lipschitz continuity. This is still sufficient for our argument, since we only need continuity. A milder version of the Schr\u00f6dinger equation would allow for more general potentials in some $L^p$ spaces in time~\\cite{PenzRuggenthaler2015, RuggenthalerPenzLeeuwen2015}. In that case, however, potentials which only differ at a set of zero measure would be considered equivalent.}, so `almost everywhere' could be dropped. For more details on the solvability of the time-dependent Schr\u00f6dinger equation, I refer the reader to an excellent introduction in Ref.~\\cite{RuggenthalerPenzLeeuwen2015}. This first possibility is trivial, and will be excluded from further discussion.\nThe other possibility is that there exists one or more linear combinations of the operators under consideration\n\\begin{align*}\n\\hat{L}_n = \\sum_j\\hat{Q}_j\\delta v_j^n,\n\\end{align*}\nsuch that\n$\\brakket{\\Psi_0}{\\hat{L}_n}{\\Psi_K} = 0$ for all $K \\geq D$. Note that the linear combination should remain hermitian, so $\\delta v_j^n \\in \\mathbb{R}$.\nThis implies that such a linear combination acting on the initial state, $\\hat{L}_n\\ket{\\Psi_0}$, should not produce any components outside the degenerate subspace, i.e.\n\\begin{align}\\label{eq:f0requirement}\n\\hat{L}_n\\ket{\\Psi_0} = \\sum_{K < D}{l_n^K}^*\\ket{\\Psi_K} ,\n\\end{align}\nwhere $l^K_n \\coloneqq \\brakket{\\Psi_0}{\\hat{L}_n}{\\Psi_K}$.\nIn the case of a non-degenerate ground state the situation simplifies to an eigenvalue condition\n\\begin{align}\\label{eq:eigenCond}\n\\hat{L}_n\\ket{\\Psi_0} = l_n\\ket{\\Psi_0} .\n\\end{align}\nIn words, for a non-degenerate initial ground state, the response can only be zero nontrivially, if there exists a linear combination of the operators $\\hat{Q}_j$ for which the initial state is an eigenstate. Note $\\ket{\\Psi_0}$ being an eigenstate of $\\hat{L}_n$ is sufficient, though not necessary for degenerate ground states, since $\\hat{L}_n\\ket{\\Psi_0}$ is still allowed to have components in the degenerate subspace~\\eqref{eq:f0requirement}.\n\nThough we have shown that $a_K(t) = 0$ for $K \\geq D$ is necessary for absence of response, we also need to check if this condition is sufficient.\nNow suppose that condition~\\eqref{eq:f0requirement} holds for some initial state $\\ket{\\Psi_0}$ and some operator $\\hat{Q}_n$.\nNote that we can always make a linear transformation of the set of operators such that the operators $\\hat{L}_n$ are all explicitly contained in the set $\\{\\hat{Q}_n\\}$.\nIn that case only the states with $\\Omega_K = 0$ will contribute to the sum-over-state expression in~\\eqref{eq:sosRespFunc}, so reduces to\n\\begin{align*}\n\\chi_{ni}(t-t') &=\n\\mathrm{i}\\theta(t-t')\\sum_K \\Imag\\bigl[{q_n^K}^*q_i^K\\bigr] +\\text{c.c.} \\\\\n&= \\mathrm{i}\\theta(t-t')\\brakket{\\Psi_0}{[\\hat{Q}_i,\\hat{Q}_n]}{\\Psi_0} .\n\\end{align*}\nso as an additional requirement for zero response apart from $q^K_n = 0$ for $K \\geq D$, we find that\n\\begin{align}\\label{eq:fDegenComm}\n\\brakket{\\Psi_0}{[\\hat{Q}_i,\\hat{Q}_n]}{\\Psi_0} = 0 \\qquad \\forall_i .\n\\end{align}\nA number of remarks on this condition are in order. If the initial state is an eigenstate of the operator $\\hat{Q}_n$, this condition is automatically satisfied. Hence, it is sufficient for non-degenerate ground states to check condition~\\eqref{eq:eigenCond}.\nThis implies that only in the case of a degenerate initial state for which $\\hat{Q}_n\\ket{\\Psi_0}$ has some components in the degenerate subspace, condition~\\eqref{eq:fDegenComm} needs to considered explicitly. Since operators commute with themselves, condition~\\eqref{eq:fDegenComm} is trivially satisfied for $i=n$, so the check only needs to be performed for $i \\neq n$.\n\n\\textit{Example.}\nTo get a feeling how condition~\\eqref{eq:fDegenComm} comes into play, consider a three dimensional Hilbert space, $\\mathcal{H} = \\{\\ket{2s},\\ket{2p_x},\\ket{2p_y}\\}$.\nWe take the Hamiltonian of the hydrogen atom as our initial Hamiltonian, $\\hat{H}_0$, so the states in $\\mathcal{H}$ are degenerate.\nThe condition~\\eqref{eq:f0requirement} is therefore trivially satisfied and only the additional condition~\\eqref{eq:fDegenComm} needs to be considered.\nFor the operators $\\hat{Q}_j : \\mathcal{H} \\to \\mathcal{H}$ we consider the unit operator and the dipole operators in the $x$- and $y$-direction, $\\{\\hat{Q}_j\\} = \\{\\mathbbm{1}, x, y\\}$.\nWe select the $2s$-orbital to be our initial state, $\\ket{\\Psi_0} = \\ket{2s}$.\nSince any state is an eigenstate of the unit operator, we immediately find that at least the unit operator is a part of the kernel of the response function. The dipole operators produce an additional component in the degenerate subspace~\\footnote{The matrix elements {$\\ket{\\psi_i}\\brakket{\\psi_i}{\\coord{r}}{2s}$} have been evaluated by calculating the corresponding integral.}\n\\begin{align*}\nx\\ket{2s} &\\coloneqq \\sum_{\\crampedclap{\\psi_i \\in \\mathcal{H}}}\\ket{\\psi_i}\\brakket{\\psi_i}{x}{2s} = -3\\ket{2p_x} ,\t\\\\\ny\\ket{2s} &\\coloneqq \\sum_{\\crampedclap{\\psi_i \\in \\mathcal{H}}}\\ket{\\psi_i}\\brakket{\\psi_i}{y}{2s} = -3\\ket{2p_y} ,\n\\end{align*}\nso the additional condition~\\eqref{eq:fDegenComm} needs to be checked explicitly. The dipole operators already commute among themselves, $[x,y] = 0$, so also the dipole operators belong to the kernel of the response function. Because all operators belong to the kernel of the response function, we actually have $\\mat{\\chi} = \\mat{0}$. Such a response function is rarely encountered in practice and is a consequence of the special choice of the Hilbert space, $\\mathcal{H}$, and the set of operators $\\{\\mathbbm{1}, x, y\\}$. If, for example, also the $\\ket{3p_x}$ state would be included in the Hilbert space, the $x$-operator would produce also components outside the degenerate subspace\n\\begin{align*}\nx\\ket{2s} = \\sum_{\\crampedclap{\\psi_i \\in \\mathcal{H} \\cup \\{\\ket{3p_x}\\}}}\\ket{\\psi_i}\\brakket{\\psi_i}{x}{2s}\n=  \\frac{27\\,648}{15\\,625}\\ket{3p_x} - 3\\ket{2p_x} .\n\\end{align*}\nBy the first condition~\\eqref{eq:f0requirement}, the $x$-operator would not belong to the kernel of the response function anymore. Note that the extension has only consequences for the $x$-operator, so the unit-operator and $y$-operator remain in the kernel of the response function.\n\nThe $x$-operator can also be lifted out of the kernel of the response function by adding additional operators. For example, consider an extension of the set of operators by an operator which applies the momentum operator three times in the $x$-direction, $\\mathrm{i}\\partial_x^3 = (-\\mathrm{i}\\partial_x)^3$~\\footnote{The momentum operator gives {$-\\mathrm{i}\\partial_x\\ket{2s} = 0$} in the limited Hilbert space $\\mathcal{H}$, so is simply the zero-operator. Using $(-\\mathrm{i}\\partial_x)^3$ avoids such a pathological operator.}. The action of $\\mathrm{i}\\partial_x^3$ on the initial state is defined to be\n\\begin{align*}\n\\mathrm{i}\\partial_x^3\\ket{2s} \\coloneqq \\sum_{\\crampedclap{\\psi_i \\in \\mathcal{H}}}\\ket{\\psi_i}\\brakket{\\psi_i}{\\mathrm{i}\\partial_x^3}{2s}\n= \\frac{\\mathrm{i}}{20}\\ket{2p_x} .\n\\end{align*}\nConsider now the additional condition~\\eqref{eq:fDegenComm} for $\\mathrm{i}\\partial_x^3$. Only the dipole operator in the $x$-direction yields a non-vanishing commutator and its expectation value for the initial state gives\n\\begin{align*}\n\\bigbrakket{2s}{\\bigl[x,\\mathrm{i}\\partial_x^3\\bigr]}{2s} = \\frac{\\mathrm{i}}{4} \\neq 0 .\n\\end{align*}\nThe operator $\\mathrm{i}\\partial_x^3$ is therefore not part of the kernel of the `extended' response function. Note that this result also implies that for the dipole operator in the $x$-direction condition~\\eqref{eq:fDegenComm} is not satisfied anymore. Only the unit operator and the dipole operator in the $y$-direction span therefore the kernel of the `extended' response function.\n\n\n\n\n\n\\section{Density response}\nAs a minor check, let us consider the density response function to see if we recover the original result by Van Leeuwen~\\cite{Leeuwen2001}. For the density response function our operators are $\\hat{Q}_{\\coord{r}} = \\hat{n}(\\coord{r})$, where $\\hat{n}(\\coord{r}) \\coloneqq \\sum_{\\sigma}\\crea{\\psi}(\\coord{r}\\sigma)\\anni{\\psi}(\\coord{r}\\sigma)$. The only linear combination for which a non-degenerate ground state is an eigenstate is the number operator\n\\begin{align}\\label{eq:totalNumberOperators}\n\\hat{N} \\coloneqq \\integ{\\coord{r}}\\hat{n}(\\coord{r}) .\n\\end{align}\nOnly if the density would vanish in some region for $\\ket{\\Psi_0}$, there would be other linear combinations for which $\\ket{\\Psi_0}$ would be an eigenstate. This possibility is typically excluded in DFT~\\cite{HohenbergKohn1964, ReedSimon1980, Lieb1983, Lammert2015}, so we recover the same result as Van Leeuwen~\\cite{Leeuwen2001} that only a spatially constant potential gives a zero density response.\n\nNow let us investigate the consequences of a degenerate ground state by considering only one particle first. A non-constant potential yielding a zero density response is readily constructed as \n$v_K(\\coord{r}) = \\Psi_K(\\coord{r}) \/ \\Psi_0(\\coord{r})$ for $0 < K < D$,\nwhich by construction satisfies~\\eqref{eq:f0requirement}. This construction only works if the wave function $\\Psi_K(\\coord{r})$ has no imaginary part. Otherwise the potential $v_K$ would not be hermitian.\nAssuming that the ground states $\\Psi_K(\\coord{r})$ are indeed real, the initial state should at least have one nodal surface to allow for the degeneracy. Further, because the states need to be orthogonal, not all of their nodal surfaces should coincide. The potential $v_K$ would therefore be infinite along some nodal surface of $\\Psi_0$~\\footnote{An example is the hydrogen atom where the $1s$ state is excluded from the Hilbert space. Choose the $2p_x$ orbital as an initial state. The local potential which would produce only a component in the $2p_y$ state would be $y\/x$ which is infinite in the whole plane orthogonal to the $x$-axis.}.\n\nFor the next step we need to take additional conditions on the allowed perturbation into account, to ensure that the Hamiltonian remains self-adjoint~\\cite{ReedSimon1975, RuggenthalerPenzLeeuwen2015}. For potentials over $\\mathbb{R}^3$ one requires the perturbations to be in the class of Kato potentials $\\mathcal{K} \\coloneqq L^2(\\mathbb{R}^3) + L^{\\infty}(\\mathbb{R}^3)$~\\cite{Kato1957}. This means that the perturbing potential needs to be decomposable in parts which are either bound or square (Lebesgue) integrable. For example the Coulomb potential can be split into two parts as\n\\begin{align*}\n\\frac{1}{\\abs{\\coord{r}}} = \\frac{\\theta(1 - \\abs{\\coord{r}})}{\\abs{\\coord{r}}} + \\frac{\\theta(\\abs{\\coord{r}} - 1)}{\\abs{\\coord{r}}} .\n\\end{align*}\nThe first part contains the Coulomb singularity, which that is square-integrable, so the first part belongs to $L^2$. The second part is the outer region of the Coulomb potential which is bounded (between 0 and 1), so belongs to $L^{\\infty}$. The Coulomb potential is therefore a proper perturbative potential, since it is a Kato perturbation. An example of a potential which is not Kato is the harmonic oscillator potential, $\\frac{1}{2}\\omega^2\\abs{\\coord{r}}^2$. Due to its divergence for $\\abs{\\coord{r}} \\to \\infty$ the outer part of the harmonic potential is neither bound nor square integrable.\nReturning to our potentials $v_K(\\coord{r}) = \\Psi_K(\\coord{r}) \/ \\Psi_0(\\coord{r})$, we expect these potentials to behave as $1\/z$ in the direction orthogonal to the nodal surface. These potentials are therefore expected not to be square integrable in regions including some part of the nodal surface. To proof this suspicion, we use the result by Kato~\\cite{Kato1957} that the solutions of the time-independent Schr\u00f6dinger equation are continuous over whole $\\mathbb{R}^{3N}$ and their derivatives locally in $L^{\\infty}$, i.e.\\ the solutions are locally Lipschitz. Locally Lipschitz means that for each $\\coord{x}_0,\\coord{y}_0 \\in \\mathbb{R}^{3N}$ there exists some neighborhood $V$ and a constant $K_V$ such that\n$\\abs{\\Psi(\\coord{x}) - \\Psi(\\coord{y})} \\leq K_V\\abs{\\coord{x} - \\coord{y}}$ for any $\\coord{x},\\coord{y} \\in V$. Now take a neighborhood around some point at the nodal surface of $\\Psi_0(\\coord{r})$ and call this neighborhood $S$. The local Lipschitz condition for $\\Psi_0$ simplifies in this neighborhood to $\\abs{\\Psi_0(s)} \\leq K_S\\abs{s}$ where $s$ denotes the distance from the nodal surface. The sign of $s$ indicates on which side of the surface we are. Since the nodal surfaces of $\\Psi_K(\\coord{r})$ and $\\Psi_0(\\coord{r})$ do not coincide, we can always choose some neighborhood such that $\\min(\\abs{\\Psi_K}) = E > 0$. The $L^2$-norm of the potential within $S$ can now be estimated as\n\\begin{align*}\n\\binteg{\\coord{r}}{S}{}\\biggl\\lvert\\frac{\\Psi_K(\\coord{r})}{\\Psi_0(\\coord{r})}\\biggr\\rvert^{\\mathrlap{2}}\n&\\geq \\biggl(\\frac{E}{K_S}\\biggr)^{\\mathclap{2}}\\binteg{\\coord{r}}{S}{}\\frac{1}{s^2} \\\\\n&\\geq \\biggl(\\frac{E}{K_S}\\biggr)^{\\mathclap{2}}\\area(S^*)\\binteg{s}{-\\epsilon}{\\epsilon}\\frac{1}{s^2} = \\infty ,\n\\end{align*}\nwhere $\\area(S^*) > 0$ and $\\epsilon > 0$.\nIn the last inequality, $S^*$ is a part of the nodal surface contained in $S$, such that $S^* \\times [-\\epsilon,\\epsilon] \\subseteq S$. Thus the last integral is basically over a slab of thickness $2\\epsilon$ contained in $S$ along the nodal surface. The inequality therefore shows that the potential $v_K(\\coord{r})$ is not square integrable in the region $S$, so this part of the potential is not in $L^2$. Since the potential is obviously not bounded in this region, i.e.\\ not in $L^{\\infty}$, the potential $v_K(\\coord{r})$ is not a Kato perturbations. The potentials $v_K(\\coord{r})$ are therefore not admissible, so degenerate initial states do not form any complication for the invertibility of the density response function. The final result is that also in the degenerate case the kernel of the density response function only consists of a constant potential. It is obvious that the same conclusion also holds for more than one particle.\n\n\n\n\n\n\n\n\n\n\\section{1RDM response}\nWe will now consider the 1RDM response function. The 1RDM operator is defined as\n\\begin{align*}\n\\hat{\\gamma}(\\coord{x},\\coord{x}') \\coloneqq \\crea{\\psi}(\\coord{x}')\\anni{\\psi}(\\coord{x}) ,\n\\end{align*}\nwhere $\\anni{\\psi}(\\coord{x})$ and $\\crea{\\psi}(\\coord{x})$ are the usual field operators and $\\coord{x} \\coloneqq \\coord{r}\\sigma$ is a combined space-spin coordinate.\nWe will first limit ourselves to a non-degenerate ground state as initial state, since this case already leads to several situations which need to be considered.\n\n\n\n\\subsection{Non-degenerate ground state as initial state}\n\\label{sec:nondegenerate1RDM}\n\nBecause the density is simply the diagonal of the 1RDM, $n(\\coord{r}) = \\sum_{\\sigma}\\gamma(\\coord{r}\\sigma,\\coord{r}\\sigma)$, the constant potential is also present in the kernel of the 1RDM response function.\nSince the 1RDM contains more flexibility than the density, one would expect that there are more possible potentials that give a zero response than only the spatially constant potential. Indeed, any one-body operator can be represented by the 1RDM, so if the initial state is an eigenfunction of some one-body operator, this operator is also present in the kernel of the 1RDM response function.\n\nIn particular, the non-relativistic Hamiltonian does not depend on spin, so a non-degenerate ground state is necessarily a singlet state. This implies that the ground state is an eigenstate of the total spin-projection operator in arbitrary directions, $\\hat{\\mat{S}} \\ket{\\Psi_0} = 0$. Since the total spin-projection operator can be expressed as a one-body operator, it is also part of the kernel of the 1RDM response function. Note that this situation also occurs in spin-DFT~\\cite{BarthHedin1972, EschrigPickett2001}.\n\nSince symmetry in the system implies that the Hamiltonian commutes with one or more symmetry operators, the eigenstates of the Hamiltonian can be chosen to be eigenstates of some of those symmetry operators as well. Therefore, one would expect that also these symmetry operators belong to the kernel of the 1RDM response function. However, the Coulomb interaction of the Hamiltonian couples all the particles, so these symmetry operators need to be many-body operators in general. Continuous symmetries form an exception, since their generators can be expressed as one-body operators. For linear molecules this would be the rotation around the $z$-axis, i.e.\\ the $\\hat{L}_z$ operator. Atoms would also include the other total angular momentum operators, $\\hat{L}_x$ and $\\hat{L}_y$. For systems which are homogeneous in one or more directions, e.g.\\ the homogeneous electron gas, the corresponding momentum operator(s) would also be part of the kernel of the 1RDM response function.\n\nTo proceed with the analysis, we will work in the natural orbital (NO) basis of the 1RDM of the initial ground state, which can be obtained by diagonalizing the 1RDM\n\\begin{align*}\n\\gamma(\\coord{x},\\coord{x}') = \\sum_kn_k\\,\\phi_k(\\coord{x})\\phi_k^*(\\coord{x}') .\n\\end{align*}\nThe eigenvalues are called the (natural) occupation numbers and the eigenfunctions are the natural orbitals~\\cite{Lowdin1955}. The occupation numbers sum to the total number of particles in the system, $N$, and for fermions they obey $0 \\leq n_k \\leq 1$. The integer values are special, since $n_k = 0$ implies that the NO $\\phi_k(\\coord{x})$ is not present in any determinant in the expansion of the wavefunction, $\\anni{a}_k\\ket{\\Psi_0} = 0$, where $\\anni{a}_k$ is the annihilation operator for the NO $\\phi_k(\\coord{x})$.\nLikewise, a fully occupied NO, $n_k = 1$, implies that the NO $\\phi_k(\\coord{x})$ is present in all determinants, so $\\crea{a}_k\\anni{a}_k\\ket{\\Psi_0} = \\ket{\\Psi_0}$~\\cite{Lowdin1955}. From these properties, we readily find that\n\\begin{align*}\n\\hat{\\gamma}_{k,l}\\ket{\\Psi_0} = \\begin{cases}\n0\\ket{\\Psi}\t\t\t\t\t\t\t&\\text{if $n_k = 0$ or} \\\\\n\t\t\t\t\t\t\t\t&\\text{\\hphantom{if }$n_l = 1$ and $k \\neq l$} \\\\\n1\\ket{\\Psi}\t\t\t\t\t\t\t&\\text{if $n_l = 1$ and $k = l$} \\\\\n{\\displaystyle \\sum_K}c_K\\ket{\\Psi_K}\t&\\text{otherwise} ,\n\\end{cases}\n\\end{align*}\nwhere the 1RDM operator is now represented in the NO basis, $\\hat{\\gamma}_{k,l} \\coloneqq \\crea{a}_l\\anni{a}_k$.\nHence we find that the ground state is an eigenstate of the 1RDM operator if $n_k = 0$ or $n_l = 1$. However, we have to keep in mind that the potential should be hermitian, so if $\\delta v_{kl} \\neq 0$, also $\\delta v^*_{lk} \\neq 0$. Thus for the state $\\ket{\\Psi}$ to be an eigenstate of both $\\hat{\\gamma}_{k,l}$ and $\\hat{\\gamma}_{l,k}$, we additionally need that $n_k = 1$ or $n_l = 0$. This situation can only occur if $n_k = n_l = 0$ or $n_k = n_l = 1$. We find therefore, that the perturbations within the fully occupied block or within the completely unoccupied block have a zero response in the 1RDM, as is actually well known for non-interacting systems, e.g.\\ the Kohn--Sham system in DFT.\nNote that this discussion includes the one-particle case, since that is also non-interacting.\n\n\nFor interacting systems, the occupation numbers are predominantly fractional, $0 < n_k < 1$, and for Coulomb systems there is strong evidence that they all are~\\cite{Friesecke2003, GiesbertzLeeuwen2013a, GiesbertzLeeuwen2013b, GiesbertzLeeuwen2014}. One would expect that another special situation can occur if these fractional occupation numbers are degenerate. To investigate this situation, consider the NOs as a basis and assume that $\\phi_1(\\coord{x})$ and $\\phi_2(\\coord{x})$ are two degenerate NOs. The contribution of these degenerate NOs to the initial state can be made explicit by writing the initial state as\n\\begin{align*}\n\\ket{\\Psi_0} = \\crea{a}_1\\crea{a}_2\\ket{\\widetilde{\\Psi}^{12}_{N-2}} + \n\\crea{a}_1\\ket{\\widetilde{\\Psi}^1_{N-1}} +\n\\crea{a}_2\\ket{\\widetilde{\\Psi}^2_{N-1}} + \\ket{\\widetilde{\\Psi}_N} ,\n\\end{align*}\nwhere $\\anni{a}_i\\ket{\\widetilde{\\Psi}^b_M} = 0$ for $i = 1,2$ and any $b \\in \\{\\emptyset, 1, 2, 12\\}$.\nNote that the states $\\ket{\\widetilde{\\Psi}^a_M}$ are not normalized in general.\nThe action of the 1RDM-operator on the initial state can be worked out as\n\\begin{align*}\n\\hat{\\gamma}_{i,j}\\ket{\\Psi_0}\n= \\crea{a}_j\\ket{\\widetilde{\\Psi}^i_{N-1}} + \\delta_{i,j}\\crea{a}_1\\crea{a}_2\\ket{\\widetilde{\\Psi}^{12}_{N-2}}\n\\end{align*}\nfor $i,j = 1,2$. Since the $\\ket{\\widetilde{\\Psi}_N}$ component vanishes, the only way that $\\ket{\\Psi_0}$ can be an eigenstate is to have the eigenvalue zero, so all components $\\ket{\\widetilde{\\Psi}^b_M}$ need to be cancelled. The components $\\crea{a}_j\\ket{\\widetilde{\\Psi}^i_{N-1}}$ are not present in the initial state $\\ket{\\Psi_0}$. This follows from the fact that $\\phi_1(\\coord{x})$ and $\\phi_2(\\coord{x})$ are NOs, so $\\gamma_{12} = 0$. Since one generally can not eliminate these components by taking linear combinations of $\\hat{\\gamma}_{i,j}$, fractional occupation degeneracies do \\emph{not} cause additional potentials in the kernel of the 1RDM response function in general.\n\nA special situation occurs if $\\ket{\\widetilde{\\Psi}^i_{N-1}} = 0$. The only known interacting case is the two-electron system. The two-electron state in the NO representation can be written as an expansion of NO pairs to which each NO contributes only once~\\cite{LowdinShull1956, CioslowskiPernalZiesche2002, PhD-Giesbertz2010, RappBricsBauer2014}\n\\begin{align*}\n\\ket{\\Psi_0} = \\sum_{k=1}^{\\infty}c_k\\,\\crea{a}_{\\vphantom{\\bar{k}}k}\\crea{a}_{\\bar{k}}\\ket{} .\n\\end{align*}\nThe coefficients in the expansion are called natural amplitudes and are related to the occupation numbers as $\\abs{c_k}^2 = n_k = n_{\\bar{k}}$. In the case of a singlet state, the NO pairs only differ in their spin part, $\\phi_k(\\coord{x}) = \\phi_k(\\coord{r})\\alpha(\\sigma)$ and $\\phi_{\\bar{k}}(\\coord{x}) = \\phi_k(\\coord{r})\\beta(\\sigma)$. In the triplet case the NO pairs have different spatial parts and their spin parts can be identical. The paired NOs are degenerate and since we now have $\\ket{\\widetilde{\\Psi}^k_{N-1}} = 0$, we find that\n\\begin{align}\\label{eq:pairedZeroResponse}\n0 = \\hat{\\gamma}_{k,\\bar{k}}\\ket{\\Psi_0} = \\hat{\\gamma}_{\\bar{k},k}\\ket{\\Psi_0}\n= \\bigl(\\hat{\\gamma}_{k,k} - \\hat{\\gamma}_{\\bar{k},\\bar{k}}\\bigr)\\ket{\\Psi_0} ,\n\\end{align}\nso perturbations with these operators yield a zero 1RDM response~\\footnote{If $\\Psi_0$ is a singlet state,~\\eqref{eq:pairedZeroResponse} are triplet operators.}.\n\nThe special structure of the two-electron state also causes other NOs with degenerate natural occupation numbers to yield zero response. For example, consider the contribution of two pairs of NOs to the initial state\n\\begin{align*}\nc_1\\crea{a}_{\\vphantom{\\bar{1}}1}\\crea{a}_{\\bar{1}}\\ket{} + c_2\\crea{a}_{\\vphantom{\\bar{2}}2}\\crea{a}_{\\bar{2}}\\ket{} .\n\\end{align*}\nNow we work out the action of the following perturbations which mix these NO pairs\n\\begin{align*}\n\\bigl(v_{21}\\hat{\\gamma}_{1,2} + v_{21}^*\\hat{\\gamma}_{2,1}\\bigr)\\ket{\\Psi_0}\n&= v_{21}c_1\\crea{a}_{\\vphantom{\\bar{1}}2}\\crea{a}_{\\bar{1}}\\ket{} +\nv_{21}^*c_2\\crea{a}_{\\vphantom{\\bar{2}}1}\\crea{a}_{\\bar{2}}\\ket{} , \\\\\n\\bigl(v_{\\bar{1}\\bar{2}}\\hat{\\gamma}_{\\bar{2},\\bar{1}} + v_{\\bar{1}\\bar{2}}^*\\hat{\\gamma}_{\\bar{1},\\bar{2}}\\bigr)\\ket{\\Psi_0}\n&= v_{\\bar{1}\\bar{2}}\\,c_2\\crea{a}_{\\vphantom{\\bar{1}}2}\\crea{a}_{\\bar{1}}\\ket{} + \nv_{\\bar{1}\\bar{2}}^*\\,c_1\\crea{a}_{\\vphantom{\\bar{2}}1}\\crea{a}_{\\bar{2}}\\ket{} .\n\\end{align*}\nNote that we added a second term to each perturbation, to ensure that the operator is hermitian.\nWe see that both perturbations produce exactly the same determinants, so by combining both perturbations we might be able to cancel both. To eliminate the first determinant, $\\crea{a}_2\\crea{a}_{\\bar{1}}\\ket{}$, we need to set $v_{\\bar{1}\\bar{2}} = -v_{21}c_1\/c_2$. To eliminate the second determinant, $\\crea{a}_1\\crea{a}_{\\bar{2}}\\ket{}$, we need to set $v^*_{\\bar{1}\\bar{2}} = -v^*_{21}c_2\/c_1$. This only works when the natural occupations are degenerate, $n_1 = \\abs{c_1}^2 = \\abs{c_2}^2 = n_2$. In that case the following potential belongs to the kernel of the 1RDM response function\n\\begin{align}\\label{eq:kernPotDegen1}\nv_{21}\\bigl(\\hat{\\gamma}_{1,2} - \\textrm{e}^{\\mathrm{i}\\alpha_{12}} \\gamma_{\\bar{2},\\bar{1}}\\bigr) +\nv_{21}^*\\bigl(\\hat{\\gamma}_{2,1} - \\textrm{e}^{-\\mathrm{i}\\alpha_{12}} \\gamma_{\\bar{1},\\bar{2}}\\bigr) ,\n\\end{align}\nwhich depends on the relative phase of the natural amplitudes, $\\textrm{e}^{\\mathrm{i}\\alpha_{12}} \\coloneqq c_1\/c_2$.\nIt is readily checked that degeneracy between the NO pairs implies that also the potential\n\\begin{align}\\label{eq:kernPotDegen2}\nv_{\\bar{2}1}\\bigl(\\hat{\\gamma}_{1\\bar{2}} + \\textrm{e}^{\\mathrm{i}\\alpha_{12}} \\hat{\\gamma}_{2\\bar{1}}\\bigr) +\nv_{\\bar{2}1}^*\\bigl(\\hat{\\gamma}_{\\bar{2}1} + \\textrm{e}^{-\\mathrm{i}\\alpha_{12}} \\hat{\\gamma}_{\\bar{1}2}\\bigr)\n\\end{align}\nbelongs to the kernel of the response function. Note that the relative phase of the natural amplitude is important in the construction of the potentials~\\eqref{eq:kernPotDegen1} and~\\eqref{eq:kernPotDegen2}. Since the natural amplitudes can only be defined for a two-electron system, the notion of relative phases only makes sense for two-electron systems. The relative phases therefore emphasize the special status of the interacting two-electron system concerning NO degeneracies.\n\nAll non-local one-body potentials in the kernel of the 1RDM response function have now been characterized for both the non-interacting case and the fully interacting Coulomb system. However, if one works in a small finite basis or uses some effective interaction which only affects some subspace, some special structure in the ground state might arise which causes additional potentials to be present in the kernel of the 1RDM response function. A complete proof including these cases would therefore require additional assumptions or a more extensive analysis which would depend on the specific details.\n\n\n\n\\subsection{Including degeneracies}\n\nNow let us consider if additional potentials will be part of the kernel of the 1RDM response function if we allow for degenerate ground states. We will do this by first assuming that there exists some non-local one-body potential which only creates components in the degenerate subspace when acting on the initial state\n\\begin{align}\\label{eq:UpotDef}\n\\hat{U}\\ket{\\Psi_0} = \\sum_{i,j}u_{ji}\\hat{\\gamma}_{i,j}\\ket{\\Psi_0}\n= \\sum_{\\mathclap{0 \\leq K < D}}u^*_K\\ket{\\Psi_K} \\neq 0 ,\n\\end{align}\nwhere $u_K = \\brakket{\\Psi_0}{\\hat{U}}{\\Psi_K}$.\nSubsequently we check whether the additional necessary condition~\\eqref{eq:fDegenComm} is satisfied. This condition attains the following simple form in the NO basis\n\\begin{align}\\label{eq:fDegen1RDM}\n0 = \\bigbrakket{\\Psi_0}{\\bigl[\\hat{\\gamma}_{kl},\\hat{U}\\bigr]}{\\Psi_0} = (n_l - n_k)u_{kl} \\qquad \\forall_{k,l} .\n\\end{align}\nThis is a very interesting expression, since it tells us that only potentials $\\hat{U}$ which have only non-zero matrix elements coupling degenerate NOs yield a zero 1RDM response.\nThis is a very stringent condition, especially in combination with the requirement that $\\hat{U}\\ket{\\Psi_0}$ is only allowed to have components in the degenerate subspace of ground states.\n\nNow let us investigate a number of systems of interest. First consider a system of non-interacting particles with degenerate ground states. These ground states can be constructed by first solving effective one-particle Schr\u00f6dinger equations $\\hat{h}\\ket{\\phi_i} = \\epsilon_i\\ket{\\phi_i}$. Assuming that the orbital energies are ordered in increasing order, $\\epsilon_i \\leq \\epsilon_{i+1}$, the complete span of degenerate ground states can be constructed by assembling all determinant with the lowest orbital energies\n\\begin{align*}\n\\ket{\\Psi_{i_1,\\dotsc,i_{N-k}}} = \\crea{a}_{i_1}\\dotsb\\crea{a}_{i_{N-k}}\\crea{a}_k\\dotsb\\crea{a}_1\\ket{} ,\n\\end{align*}\nwhere $i_1 < i_2 < \\dotsb < i_{N-k} \\in \\mathcal{D} \\coloneqq \\{k+1,k+2,\\dotsc,k+d\\}$ and $d$ denotes the number of degenerate orbitals with orbital energy $\\epsilon_{k+1} = \\epsilon_{k+2} = \\dotsb = \\epsilon_{k+d}$. Note that this set of degenerate ground states is not unique. Arbitrary unitary transformations among the degenerate ground states yield different spans of the ground state subspace which are equally valid. In particular, the initial ground state of the response function can be chosen as any linear combination of the degenerate ground states.\n\nA potential $\\hat{U}$ satisfying condition~\\eqref{eq:UpotDef} is readily constructed by setting $u_{ij} = 0$ if any $i,j \\notin \\mathcal{D}$ and choosing some $u_{i,j} \\neq 0$ for both $i,j \\in \\mathcal{D}$. for the potential $\\hat{U}$ to belong to the kernel of the 1RDM response function, we additionally need that $n_i = n_j$ for all elements $u_{i,j} \\neq 0$. Note that the cases $n_i = n_j = 1$ and $n_i = n_j = 0$ were already covered before in the non-degenerate case, since $\\ket{\\Psi_0}$ will be actually an eigenstate of the potential $\\hat{U}$. The treatment of degenerate non-interacting ground states extends this result to any potential coupling only degenerate NOs. So all potentials with $u_{ij} = 0$ for $n_i \\neq n_j$ will belong to the kernel of the 1RDM response function in the non-interacting case.\n\nNow let us consider a system with a spin degenerate ground states, $\\ket{S,M}$. These states are eigenfunctions of the $\\hat{S}_z$ operator, $\\hat{S}_z\\ket{S,M} = M\\ket{S,M}$, so the $\\hat{S}_z$ operator immediately belongs to kernel of the 1RDM response function. By operating with the $\\hat{S}_{\\pm}$ operators we can obtain other states in the degenerate subspace, $\\hat{S}_{\\pm}\\ket{S,M} = C_{\\pm}(S,M)\\ket{S,M\\pm1}$. In second quantizations, these raising and lowering operators can be expressed as\n\\begin{align*}\n\\hat{S}_+ &= \\sum_k\\crea{a}_{k\\alpha}\\anni{a}_{k\\beta}\t&\n&\\text{and}\t&\n\\hat{S}_- &= \\sum_k\\crea{a}_{k\\beta}\\anni{a}_{k\\alpha} .\n\\end{align*}\nThe operators $\\hat{S}_{\\pm}$ are not hermitian operators, but we can make two independent hermitian combinations which are properly hermitian\n\\begin{align*}\n\\hat{S}_x &= \\frac{1}{2}\\bigl(\\hat{S}_+ + \\hat{S}_-\\bigr)\t&\n&\\text{and}\t&\n\\hat{S}_y &= \\frac{1}{2\\mathrm{i}}\\bigl(\\hat{S}_+ - \\hat{S}_-\\bigr) .\n\\end{align*}\nSince these operators only produce components in the degenerate subspace, we have found proper potentials $\\hat{U}$ as in~\\eqref{eq:UpotDef}. Now we need to check whether these operators satisfy~\\eqref{eq:fDegen1RDM}. We see that the $\\hat{S}_x$ and $\\hat{S}_y$ operators couple the different spin components of each spatial orbital, so we need $n_{k\\alpha} = n_{k\\beta}$ for all $k$ for~\\eqref{eq:fDegen1RDM} to hold. This degeneracy only occurs for $M = 0$, so only in the case that $\\ket{S,0}$ is the initial state, the $\\hat{S}_x$ and $\\hat{S}_y$ operators also belong to the kernel of the 1RDM response function. Combined with our result for non-degenerate states, this means the $\\hat{S}_z$ is always part of the the kernel of the 1RDM response function for Hamiltonians not depending on spin. If additionally the system is spin-compensated, i.e.\\ $n_{k\\alpha} = n_{k\\beta}$ for all $k$, the $\\hat{S}_x$ and $\\hat{S}_y$ operators are also part of the kernel, irrespective if the ground state is degenerate or not. Note that the same considerations also hold for the angular momentum operators $\\hat{\\mat{L}}$ if the Hamiltonian is invariant under all rotations, e.g.\\ atoms and the homogeneous electron gas, though we need to check for different degeneracies in the occupation spectrum. For example consider an atom. The $z$-axis can always be chosen such that the ground state is also an eigenstate of the $\\hat{L}_z$ operator. For the $\\hat{L}_x$ and $\\hat{L}_y$ operators to be part of the kernel of the 1RDM response function as well, we need $n_{k,l,m} = n_{k,l,m'}$, which implies that the 1RDM will be unperturbed when we make rotations around an arbitrary axis.\n\n\n\n\n\n\n\n\\subsection{Ground 1RDM functional theory}\nThe generalized invertibility theorem for the non-degenerate case is equally valid for the time-\\emph{in}dependent response function by taking the $s \\to 0$ limit of the Laplace transformed quantities. The generalized invertibility theorem therefore provides the perfect opportunity to give a better classification of the uniqueness of the mapping from non-local one-body potentials to 1RDMs, $\\hat{v} \\mapsto \\gamma$. As Gilbert already mentioned in 1975~\\cite{Gilbert1975}, the class of potentials which map to the same ground state 1RDM will be larger than in DFT, but to the author's knowledge no attempt has been made to give a full classification of this non-uniqueness. We will show that the kernel of the time-dependent 1RDM response function exactly corresponds to the non-uniqueness of the non-local potential in ground 1RDM functional theory in the non-degenerate case.\n\nAs Gilbert already showed~\\cite{Gilbert1975}, the second part of the Hohenberg--Kohn theorem can straightforwardly be generalized to 1RDMs: the 1RDM of a non-degenerate ground state is unique. In other words, consider all the ground states corresponding to different non-local potentials, then there is a one-to-one correspondence between the non-degenerate ground states and their corresponding 1RDMs.\n\nNow assume that there are two (non-local) potentials, $\\hat{v}_1$ and $\\hat{v}_2$, yielding the same non-degenerate ground state. Since the Schr\u00f6dinger equation is linear, the potentials $\\hat{v}_{\\lambda} = (1-\\lambda)\\hat{v}_1 + \\lambda\\hat{v}_2$ yield exactly the same non-degenerate ground state. The set of potentials which yield the same non-degenerate ground state is therefore (simply) connected. To determine this set, it is therefore sufficient to consider a perturbation to one of these potentials and check which potentials do not lead to a response to any order. As we have shown before, the first order 1RDM response only vanishes if the ground state is an eigenstate of the perturbation, but this also immediately implies that the response will vanish to any order. We can therefore conclude that the kernel of the 1RDM response function exactly coincides with the class of potentials yielding the same ground state 1RDM. More precisely, two non-local one-body potentials yield the same ground state (1RDM) if and only if their difference is part of the kernel of the 1RDM response function. Note that these considerations are not special for the 1RDM, but can be applied to any density-functional-like theory for which we are able to characterize the kernel of the response function.\n\n\\begin{figure}[t]\n  \\includegraphics[width=\\columnwidth]{degeneracy}\n  \\caption{A sketch of the dependence of the lowest eigenstates of some quantum system as a function of a perturbation. The energy dependence of the initial state in the time-dependent case is shown by the thick line and is simply a linear function. Time-\\emph{in}dependent perturbation theory always selects the lowest perturbed state, so $\\ket{\\Psi_1}$ on the negative side and $\\ket{\\Psi_2}$ on the positive site up to the next degeneracy point. To emphasize the jumps at the degeneracy points, the lowest energies are colored red.}\n  \\label{fig:Degeneracy}\n\\end{figure}\n\nThe degenerate case is beyond the scope of this article. The main reason is that the degenerate case is handled in a fundamentally different manner in time-dependent and time-independent perturbation theory. Time-independent perturbation theory is based on the time-independent Schr\u00f6dinger equation, which is an eigenvalue equation. The ground state is therefore only specified up to its degenerate subspace, from which an appropriate $\\ket{\\Psi_0}$ needs to be chosen. This is reflected in its perturbation theory, since we need to diagonalize the perturbation in the degenerate subspace and take the lowest eigenvalue. The perturbed state therefore depends on the direction of the perturbation as illustrated by the lowest energy surface in Fig.~\\ref{fig:Degeneracy}. Time-dependent perturbation theory is based on the time-dependent Schr\u00f6dinger equation, which is an initial value problem. The initial state $\\ket{\\Psi_0}$ is therefore completely specified from the start, even in the degenerate case. Therefore, contrary to the time-independent case, we do no need to (and can not) diagonalize the perturbation in the degenerate subspace to select an appropriate zeroth order state, since it is simply specified from the start as $\\ket{\\Psi_0}$. The time-dependent situation is illustrated in Fig.~\\ref{fig:Degeneracy} by the thick line. Due to the fundamental difference in dealing with degeneracies, the result for the time-dependent response function does not straightforwardly carry over to the time-independent response function in the degenerate case and a separate treatment is required.\n\n\n\n\n\n\\section{Conclusion}\nTo summarize, I have generalized the first step of the invertibility theorem for the density response function by Van Leeuwen~\\cite{Leeuwen2001}, to arbitrary operators and to degenerate ground states. For the nontrivial absence of response, it is sufficient that initial ground state is an eigenstate of the perturbation operator and also necessary in the case of a non-degenerate ground state. For a degenerate ground state, however, the action of the perturbation operator is allowed to yield additional components in the degenerate subspace, though the expectation value of the commutator of the perturbation with any operator under consideration needs to vanish to yield zero response as an additional condition~\\eqref{eq:fDegenComm}.\n\nThe theorem can be used to establish density-functional-like theories in the time-dependent linear response regime. The restriction to ground states is not very severe, since this is the initial state which is used almost exclusively in practical linear response calculations. The determination of the kernel of the time-dependent response function also immediately carries over to the time-\\emph{in}dependent response function if the initial ground state is non-degenerate. This result is useful to establish time-independent density-functional-like theories. The kernel of the response function exactly coincides with the $v \\mapsto Q$ mapping in the non-degenerate case. The non-uniqueness of the potential can therefore be fully characterized as in the first Hohenberg--Kohn theorem for DFT.\n\nThe generalized invertibility theorem has been applied to the density response function and it has been established that only the spatially constant potential belongs to its kernel, even for a degenerate ground state. Applying the theorem to the 1RDM response function revealed that not only the constant time-dependent shift is part of the kernel, but also generators of continuous symmetries are possibly included. For non-relativistic Hamiltonians this would always be the $\\hat{S}_z$ operator and if the NOs are degenerate in both spin channels, $n_{k\\alpha} = n_{k\\beta}$, also the other components of $\\hat{\\mat{S}}$ belong to the kernel of the 1RDM response function, cf.\\ non-collinear spin-DFT~\\cite{BarthHedin1972, EschrigPickett2001}. Also the angular momentum operators are possibly included in the kernel of the 1RDM response function if the Hamiltonian is invariant under the corresponding rotations. The additional condition~\\eqref{eq:fDegenComm} requires the relevant NOs to be degenerate as well. It is obvious that when spin-orbit coupling is included, the relevant operators to be considered would be $\\hat{\\mat{J}} \\coloneqq \\hat{\\mat{L}} + \\hat{\\mat{S}}$ instead of $\\hat{\\mat{L}}$ and $\\hat{\\mat{S}}$ separately. For homogeneous systems, e.g.\\ the homogeneous electron gas, also the momentum operators $-\\mathrm{i}\\nabla$ need to be considered.\nFurther, the matrix elements of the non-local potential which couple within the fully unoccupied block or within the fully occupied block also do not lead to a first order response. \nIn the non-interacting case actually all potentials coupling only degenerate NOs belong to the kernel of the 1RDM response function.\nDue to the intimate relation between a two-electron state and its 1RDM, degeneracies of the natural occupation numbers give rise to additional non-local potentials in the kernel of the 1RDM response function, whose matrix elements couple the natural orbitals within the degenerate sub-block.\nThis result not only puts time-dependent linear response 1RDM functional theory on a rigorous basis but is also of high importance for ground state 1RDM functional theory, because it allows for a full characterization of the non-uniqueness of the non-local potential for non-degenerate ground states for the first time.\n \n\n\n\n\n\n\n\\begin{acknowledgments}\nThe author would like to thank dr.\\ M.~Ruggenthaler prof.dr.\\ R.~van Leeuwen and prof.dr.\\ E.J. Baerends for stimulating discussions and prof.dr.\\ E.K.U. Gross for pointing out that the generalized invertibility theorem also solves the non-uniqueness problem in 1RDM functional theory. Also the critical remarks by the first reviewer are much appreciated.\nSupport from the Netherlands Foundation for Research NWO (722.012.013) through a VENI grant is gratefully acknowledged.\n\\end{acknowledgments}\n\n\n\\nocite{PenzRuggenthaler2015}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s:intro}The first invariants found during the early days of\ndevelopment of the theory of invariants of differential equations\nwere all relative invariants ~\\cite{lapla, lag, brio}, and one of\nthe very first determination of absolute invariants for differential\nequations is perhaps due to Brioschi ~\\cite{brio}, who obtained them\nas a quotient of two relative invariants. It has subsequently been\nshown that in the case of linear ordinary differential equations,\nevery fundamental absolute invariant can be expressed as a rational\nfunction.\\par\n\nThese basic properties of relative invariants have led to\nsignificant progress in the study of invariants of differential\nequations and their applications, and such studies have been largely\ninfluenced by the work of Halphen ~\\cite{halph82, halph66}, and\nForsyth ~\\cite{for-inv}. However the methods used by Halphen,\nForsyth, and earlier researchers on the subject  were very intuitive\nand most often ad hoc methods requesting tedious calculations just\nfor finding a couple of invariants.\\par\n  Based on recent advances in Lie group techniques\n ~\\cite{ovsy1, ibra-nl, olvgen, ndogftc}, infinitesimal methods haven\nbeen increasingly used for the investigation of invariants of\ndifferential equations ~\\cite{ibra-par, faz,waf,ndogschw, schw,\nmelesh}. But although these infinitesimal methods provide a more\nsystematic route for the treatment of invariants of differential\nequations and transformation groups, they have been applied to the\ndetermination of relative invariants of differential equations only\nin some very rare cases ~\\cite{ibra-lap, faz}, and even in those\ncases only some very specific relative invariants were obtained in\nthe usual way as absolute invariants corresponding to partial\nstructure-preserving transformations, in which some of the variables\nin the equation are kept constant. It therefore appears that a\nnumber of interesting properties of these relative invariants have\nnot yet been uncovered.\n\\par\n\nRestricting our attention to linear ordinary differential equations,\nwe determine in this paper some properties of relative invariants,\nconsidered as semi-invariants of the full structure-preserving\ntransformations of these equations. In particular we show that every\nabsolute invariant can be expressed as a quotient of a relative\ninvariant and the fundamental semi-invariant, and we derive a\ngeneral expression for these relative invariants. We show how these\nrelative invariants can be used to obtain invariants of all orders\nvia invariant differentiation. Our approach for any explicit\ndetermination of invariants is based on the infinitesimal method\nrecently proposed in\n ~\\cite{ndogftc}, and which contrary to the former well-known method\nof ~\\cite{ibra-nl}, does not require the knowledge of the\nstructure-preserving transformations, but rather provides it.\n\n\\section{Basic properties of relative invariants}\\label{s:basic}\n Let $G$ be a Lie group of point transformations of the form\n\\begin{equation}\\label{eq:invgp}\nx= \\phi (z, w; \\tau), \\qquad y = \\psi ( z, w; \\tau),\n\\end{equation}\n where $\\tau$ denotes collectively some arbitrary\nparameters specifying the group element in $G$. Consider on the\nother hand  a family $ \\mathcal{D}$ of differential equations of the\ngeneral form\n\\begin{equation}\\label{eq:gnl}\n\\Omega(x, y_{(n)}; \\rho)=0,\n\\end{equation}\nin which $y_{(n)}$ represents the dependent variable $y=y(x)$ and\nall its derivatives up to the order $n$, and  $\\rho$ represents\ncollectively some arbitrary functions of $x,$ or arbitrary constants\nspecifying the family element in $\\mathcal{D}.$  We say that $G$ is\nthe \\emph{equivalence group} of ~\\eqref{eq:gnl} if it is the largest\ngroup of transformations that maps elements of $\\mathcal{D}$ into\nitself. In this case, the transformed equation takes the same form\n$\\Omega(z, w_{(n)}; \\theta)=0,$ in terms of the transformed\nparameter $\\theta,$ and ~\\eqref{eq:invgp} is called the\n\\emph{structure-preserving transformations} of ~\\eqref{eq:gnl}. By a\nwell known result of Lie ~\\cite{liegc}, the transformations\n ~\\eqref{eq:invgp} induces another group of transformation $G_c$\nacting on the parameter  $\\rho$ of the differential equation, and we\nshall be interested in the invariants of this group action and its\nprolongations, and which are commonly referred to as the invariants\n(or differential invariants) of ~\\eqref{eq:gnl}. As sets, $G$ and\n$G_c$ are equal, except that they act on different spaces, and both\nare often referred to as the equivalence group of ~\\eqref{eq:gnl}.\nThus both sets will often be denoted simply by $G.$\n\\par\n\n It should be noted that in ~\\eqref{eq:gnl}, $x$\nand $y$ each denote collectively all independent and dependent\nvariables, respectively, so that the equation also includes in\nparticular all partial differential equations. However, we shall be\ninterested in this paper in a linear ordinary differential equations\nof the general form\n\\begin{equation}\\label{eq:gnlin}\ny^{(n)}+ a_1 y^{(n-1)} + a_2 y^{(n-2)}+ \\dots + a_n y=0,\n\\end{equation}\nwhere $a_j= a_j(x)$ are arbitrary functions of the independent\nvariable $x.$ The structure-preserving transformations of\n~\\eqref{eq:gnlin} can be written in the form\n\\begin{equation}\\label{eq:gnlingp}\nx= \\xi (z), \\qquad y= \\eta(z) w,\n\\end{equation}\nwhere $\\xi$ and $\\eta$ are arbitrary functions. Under\n ~\\eqref{eq:gnlingp}, the transformed equation of ~\\eqref{eq:gnlin}\ntakes the form\n\\begin{equation}\\label{eq:gnlin2}\nw^{(n)} + A_1 w^{(n-1)}+ A_2 w^{(n-2)} + \\dots+ A_n w=0,\n\\end{equation}\n where the $A_j= A_j(z)$ are the new coefficients. Let $a$ denote\ncollectively the coefficients $a_j= a_j(x).$  A differential\nfunction $F= F(a, a_{(r)}),$ where $a_{(r)}$ represents as usual the\nderivative of $a$ up to a certain order $r,$ is called a\n\\emph{relative invariant} of ~\\eqref{eq:gnlin} if\n\\begin{align}\\label{eq:dfnsemi}\n F(\\tau \\cdot (a, a_{(r)}))& =\n\\mathbf{w}(\\tau)\\cdot F(a, a_{(r)}),\n\\end{align}\nfor all $a$ and $r,$ and for all $\\tau \\in G.$ In\n ~\\eqref{eq:dfnsemi}, the weight function $\\mathbf{w}=\n\\mathbf{w}(\\tau)$ must  be a character of the group $G.$ When\n$\\mathbf{w}$ is identically equal to one, the function $F$ is called\nan \\emph{absolute invariant} of ~\\eqref{eq:gnlin}. For simplicity,\nan expression of the form $F(\\tau \\cdot (a, a_{(r)}))$ like in\n ~\\eqref{eq:dfnsemi} will often be represented by $F_\\tau,$ for a\ngiven function ~$F.$\n\n\\par\nLet's assign to an expression of the form $d^k a_j\/d x^k$ the\n\\emph{weight} $j+k$. We say that a polynomial function $F$ in the\ncoefficients $a_j$ and their derivatives has weight $m$ if each of\nits terms has constant weight $m.$ By combining  a result of Forsyth\n ~\\cite{for-inv} according to which all absolute\ninvariants of ~\\eqref{eq:gnlin} can be obtained as rational\nfunctions, and certain results obtained by Halphen ~\\cite{halph66},\nwe readily obtain the following result.\n\\begin{thm}\\label{th:quotient}\nEquation ~\\eqref{eq:gnlin} has a fundamental set of absolute\ninvariants consisting of rational functions, in which every element\n$F= S_1\/S_2$ is the quotient of two relative invariants $S_1$ and\n$S_2$ of the same weight $m,$ each of which satisfies a relation of\nthe form\n\\begin{equation}\\label{eq:quotient}\nS(\\tau \\cdot (a, a_{(r)})) =\\xi(z)^{m} S(a, a_{(r)}),\\quad \\text{\nthat is } \\quad S_\\tau= \\xi(z)^m S,\n\\end{equation}\nfor every $\\tau \\in G,$ where $S$ denotes any of the invariants\n$S_1$ and $S_2.$\n\\end{thm}\nThe  invariant $F$ of Theorem ~\\ref{th:quotient} is also said to be\nof weight $m.$ In general, a relative invariant $S$ satisfying\n$S_\\tau= \\xi^r S$ for some $r \\in \\R$ is said to be of \\emph{index}\n$r.$ We shall make the result of this theorem more precise in the\nnext section.\n\\section{The fundamental relative invariant}\n\\label{s:fdamental} For a general group of transformations $G$\nacting on a manifold $M$, a semi-invariant usually refers to a\nfunction $F$ satisfying a relation similar to that specified by\n ~\\eqref{eq:dfnsemi}, and which has the simple form\n\\begin{equation}\\label{eq:dfnsemi2}\nF(g \\cdot p)= \\mathbf{w}(g) \\cdot F(p),\n\\end{equation}\nfor all $g\\in G$ and $p \\in M$ such that $g \\cdot p$ is defined, and\nwhere the function $\\mathbf{w}$ is a character of $G.$ If we let $v$\nbe a generic element in the generating system for the Lie algebra of\n$G$ and denote by $X$ the corresponding infinitesimal generator of\nthe group action, then ~\\eqref{eq:dfnsemi2} implies that\n\\begin{subequations}\\label{eq:infisemi}\n\\begin{align}\nX \\cdot F &= - \\lambda F, \\qquad \\text{where} \\\\\n  \\lambda &= d\\, \\mathbf{w}(e) (v),\n\\end{align}\n\\end{subequations}\nand where $d\\, \\mathbf{w}(e)$ denotes the differential of\n$\\mathbf{w}$ at the identity element $e$ of $G.$ A function $F$\nsatisfying ~\\eqref{eq:infisemi} is often called a\n\\emph{$\\lambda$-semi-invariant}. By a nontrivial semi-invariant, we\nshall mean a semi-invariant which is not an absolute one, and which\nin particular is not a constant function. We now establish the\nfollowing result that relates every semi-invariant to a fundamental\nset of absolute invariants.\n\\begin{thm}\\label{th:F0}\nA function $F$ defined on $M$ is a $\\lambda$-semi-invariant of $G$\nif and only if it is of the form\n\\begin{equation}\\label{eq:F0}\nF= F_0 \\Phi,\n\\end{equation}\nwhere $F_0$ is an arbitrarily chosen  nontrivial\n$\\lambda$-semi-invariant, and $\\Phi$ is an absolute invariant, that\nis $\\Phi= \\Phi(I_1, \\dots, I_s),$ where $\\set{I_1, \\dots, I_s}$ is a\nfundamental set of absolute invariants of $G.$\n\\end{thm}\n\\begin{proof}\nSuppose that in a coordinates system $\\set{u_1, \\dots, u_q}=u$ of\n$M$ the infinitesimal generator $X$ of $G$  defined as in\n ~\\eqref{eq:infisemi} has the form\n$$ X= X_1 \\,\\partial_{u_1} + \\dots + X_q \\,\\partial_{u_q},$$\nwhere $X_j= X_j(u).$ Then by a well-known result ~\\cite{sned}, the\nequivalent system of characteristic equations associated with\n ~\\eqref{eq:infisemi} is given by the sequence of $q$ equalities\n\\begin{equation}\\label{eq:charsm}\n\\frac{d u_1}{X_1}= \\frac{d u_2}{X_2} = \\dots = \\frac{d\nu_q}{X_q}=\\frac{d F}{ -\\lambda F},\n\\end{equation}\nin which the first $q-1$ equalities are the determining equations\nfor the absolute invariants. To find the general solution to this\nsystem, we can first find the integral resulting from a combination\nof the last fraction $d F\/ (- \\lambda F)$ with any suitable ones\nfrom among the first $q$ fractions. We may assume without loss of\ngenerality that such a combination is given by an equation of the\nform\n$$  \\frac{d F}{ F} = \\frac{-\\lambda d u_k}{X_k},\\qquad\n\\text{for some $k,\\; 1\\leq k \\leq q$}.$$\nThis equation clearly has separable variables, since $F$ is\nconsidered as a new independent variable added to the coordinates\nsystem $\\set{u_1, \\dots, u_q}.$  Its integral can be written in the\nform $F \\nu= C_{q+1},$ for some function $\\nu= \\nu(u).$ If we denote\nby $I_j(u)= C_j,$ for $j=1, \\dots, q-1,$ the other integrals\ncorresponding to the first $q-1$ equalities in ~\\eqref{eq:charsm},\nand in which the $C_j$ are arbitrary constants, then the $I_j$ are\nthe absolute invariants of $G$ and  the general solution of\n ~\\eqref{eq:charsm} takes the form\n$$ \\Phi_1(I_1, \\dots, I_q, F \\nu)=0, \\quad \\text{or equivalently } F \\nu=\n\\Phi (I_1, \\dots, I_q), $$\nfor some arbitrary functions $\\Phi_1$ and $\\Phi.$ To obtain the\nfunction $F_0$ of the theorem we only need to take $F_0= 1\/\\nu,$ and\nit readily follows from the Leibnitz property of the derivation $X$\nand the definition of an absolute invariant that $F_0$ is a\n$\\lambda$-semi-invariant.\n\\end{proof}\nIt should be noted that Eq. ~\\eqref{eq:F0} also holds with the same\n$F_0$ for all prolongations of the group $G.$ Since $F_0$ is\nnontrivial, it follows in particular that the set $\\set{I_1, \\dots,\nI_q, F_0}$ is functionally independent. In contrast to the case of\nabsolute invariants, not every function of semi-invariants is again\na semi-invariant, and although the set of all semi-invariants of $G$\nforms a group under functions multiplication, the sum or difference\nof two semi-invariants is not  in general a semi-invariant.\\par\n For simplicity, we let $I$ denote\ncollectively all elements in a fundamental set of absolute\ninvariants of Eq. ~\\eqref{eq:gnlin}. We shall also often denote by\n$\\omega$ an element of the form $(a, a_{(s)}),$ which can be viewed\nas an element in the $s$th-jet space determined by the independent\nvariable $x$ and the dependent variable $a$ of Eq.\n~\\eqref{eq:gnlin}.\n\\begin{cor}\\label{co:1}\nSuppose that $S_1$ is a relative invariant of Eq. ~\\eqref{eq:gnlin}\nof index $k.$ Then every relative invariant $S_2$ of order $m$ of\nthe same equation can be put into the form\n\\begin{equation}\\label{eq:gnlsemi}\nS_2= S_1^{m\/k} \\Phi,\n\\end{equation}\nfor a certain  function $\\Phi= \\Phi(I).$ In particular $S_1^m\/S_2^k$\nis an absolute invariant.\n\\end{cor}\n\\begin{proof}\nAccording to Theorem ~\\ref{th:F0}, to prove the first part of the\ncorollary,  we only need to show that $S_1^{m\/k}$ is a\n$\\lambda$-semi-invariant for the same function $\\lambda$ as $S_2,$\nand by ~\\eqref{eq:infisemi}, we only need to show that $S_1^{m\/k}$\nis a relative invariant of the same index $m$ as $S_2.$ But since\n$(S_1)_\\tau = \\xi^k S_1$ by assumption, it readily follows that\n$(S_1^{m\/k})_\\tau = \\xi^m S_1^{m\/k}.$ For the second part we notice\nthat since $S^p$ has index $r\\!\\,p$ for every relative invariant $S$\nof index $r,$ we must have $(S_1^m\/S_2^k)_\\tau = S_1^m\/ S_2^k.$\n\\end{proof}\nThe corollary says that every relative invariant of\n~\\eqref{eq:gnlin} can be expressed as a product of an arbitrarily\nchosen, but fixed relative invariant and an arbitrary function of\n$I.$ We call such a fixed relative invariant the \\emph{fundamental\nrelative invariant} of ~\\eqref{eq:gnlin}, and denote it and its\nindex by $S_0$ and $\\sigma,$ respectively. We can now give without\nany need of calculations a simple formula relating every absolute\ninvariant with the fundamental relative invariant.\n\\begin{cor}\\label{co:2}\nEq. ~\\eqref{eq:gnlin} has a fundamental set of absolute invariants\nin which every element has the form $F = S_1 \/ S_0^{m \/\\sigma}$ for\nsome relative invariant $S_1$ of index $m.$\n\\end{cor}\n\\begin{proof}\nBy Theorem ~\\ref{th:quotient}, we may write every fundamental\nabsolute invariant of index $m$ in the form $R_1\/ R_2,$ where $R_1$\nand $R_2$ are relative invariant of index $m.$ It follows from\nCorollary ~\\ref{co:1} that $R_1= S_0^{m\/ \\sigma} \\Phi_1$ and $R_2=\nS_0^{m\/ \\sigma} \\Phi_2,$  for some functions $\\Phi_1$ and $\\Phi_2.$\nThe  corollary is thus proved by  taking $S_1= S_0^{m\/ \\sigma}\n\\Phi_1 \/ \\Phi_2.$\n\\end{proof}\n\\begin{cor}\\label{co:3}\nThe weight and the index of every relative invariant coincide.\n\\end{cor}\n\\begin{proof}\nSuppose that the relative invariant $S$ has weight $\\mu$ and index\n$m$. Then $S\/ S_0^{m \/ \\sigma}$ is an absolute invariant whose\nweight is the index  of $S_0^{m\/\\sigma},$  which is $m$ by Theorem\n ~\\ref{th:quotient}. By the same theorem, this weight is the same as\nthe weight $\\mu$ of $S,$ and this proves the corollary.\n\\end{proof}\n\n\\begin{cor}\\label{co:4}\\parbox[]{1in}{$\\quad$}\n\\begin{enumerate}\n\\item[(a)] Suppose that $\\set{I_1, \\dots, I_p}$ is a fundamental set\nof absolute invariants of a certain group of equivalence\ntransformations $G,$ and that $F_0$ is a nontrivial\n$\\lambda$-semi-invariant of $G.$ Then, $\\set{F_0, F_0 I_1, \\dots,\nF_0 I_p}$ is a maximal set of functionally independent\n$\\lambda$-semi-invariants of $G.$\n\n\\item[(b)] If $I_j= S_j\/ S_0^{m_j\/\\sigma}$ are the fundamental\nabsolute invariants of Eq. ~\\eqref{eq:gnlin} for $j=1, \\dots, p,$\nwhere $m_j$ is the index of $S_j,$ then\n$$\\mathcal{S}=\n\\set{S_0^{m\/\\sigma}, S_1^{m\/m_1}, \\dots, S_p^{m\/m_p}}\n$$\nis a fundamental set of relative invariants of index $m$ of Eq.\n ~\\eqref{eq:gnlin}.\n\\end{enumerate}\n\\end{cor}\n\n\\begin{proof}\nSince $F_0$ is a nontrivial semi-invariant of $G,$ the given set\n$\\set{F_0, F_0 I_1,  \\dots, F_0 I_p}$ clearly forms a functionally\nindependent set of semi-invariants of $G,$ by Theorem ~\\ref{th:F0}.\nBy the same theorem, any other semi-invariant of $G$ has the form\n$S= F_0 \\Phi,$ where $\\Phi= \\Phi (I_1, \\dots, I_p),$ and this\nreadily proves part (a) of the corollary.\\par\nOn the other hand, every element in the given set $\\mathcal{S}$ is\nclearly a relative invariant of index $m.$ if we replace each $I_j$\nby\n$$I_j^{ m \/ m_j} = S_j^{m\/m_j} \/ S_0^{m\/\\sigma},$$\nthe resulting set formed by the $I_{j}^{m\/m_j}$ is  a fundamental\nset of absolute invariants of the same index $m$ of Eq.\n ~\\eqref{eq:gnlin}. Thus by ~\\eqref{eq:infisemi} the elements of\n$\\mathcal{S}$ are $\\lambda$-semi-invariants corresponding to the\nsame function $\\lambda.$ Therefore, part (b) of the corollary\nreadily follows from part (a).\n\\end{proof}\nIt also follows from part (b) of this corollary that\n$\\set{S_0,S_1,\\dots,  S_p}$ is a functionally independent set of\nrelative invariants, but whose elements do not have the same index.\n\n\\section{Application to the determination of invariants}\n\\label{s:appl} It is well-known, by a result of Lie, that all higher\norder differential invariants of a transformation group can be\nobtained from a generating system of lower order ones by means of\ninvariant differentiation. This often reduces the problem of\ndetermination of invariants  to finding a generating system  of\ninvariants and the invariant differential operators. To begin with,\nsuppose that we know two functionally independent absolute\ninvariants $I_0$ and $I_1$  of equation ~\\eqref{eq:gnlin}, and that\nthese are rational functions of the form\n\\begin{equation}\\label{eq:I0}\nI_0= R_0^\\sigma\/S_0^k, \\qquad I_1= S_1^\\sigma\/S_0^{m}\n\\end{equation}\nwhere $S_0$ is the fundamental relative invariant of index $\\sigma$\nalready introduced, while $R_0$ and $S_1$ are relative invariants of\nindices $k$ and $m,$ respectively. We also assume that $S_1$ is of\norder $\\mu$ as a function of the independent variable $x,$ and that\nall other relative invariants are of order at most $\\mu$ in $x.$\nThen by means of the differential operator $\\zeta D_x,$ where\n$\\zeta= 1\/I_0', \\; I_0' = d I_0\/dx$ and  $D_x= d\/dx,$ we can\ngenerate a new absolute invariant $I_1^*= \\zeta D_x(I_1),$ whose\norder is $\\mu+1$ in general. In terms of the relative invariants in\n ~\\eqref{eq:I0}, we have\n\\begin{equation}\\label{eq:I1p}\nI_1^* =  \\frac{I_1}{I_0}\\, \\frac{R_0}{S_1}\\, \\frac{(m S_1 S_0' - r\nS_0 S_1')}{(k R_0 S_0' - r S_0 R_0{\\,\\!\\! '})},\n\\end{equation}\nwhere $h' \\equiv d h \/d x$ for every function\n $h = h(x).$ It follows from Corollary ~\\ref{co:2} that to find an absolute\ninvariant of higher order $\\mu+1,$ we only need to find a relative\ninvariant of order $\\mu+1.$ We now introduce the notation\n\n\\begin{subequations}\\label{eq:varphi}\n\\begin{alignat}{2}\n\\varphi(R_1, R_2) &= m_1 R_1 R_2' - m_2 R_2 R_1', \\quad  &\\chi(R_1,\nR_2)&= \\frac{\\left[\\varphi (R_1, R_2)\\right]^{m_2}}{R_2^{m_1+\nm2+1}}\\\\\n\\varphi_0(R_1, R_2) &= R_1, \\quad  &\\chi_0(R_1, R_2) &=\nR_1^{m_2}\/R_2^{m_1}\n\\end{alignat}\n\\end{subequations}\nfor any relative invariants $R_1$ and $R_2$ of respective index\n$m_1$ and $m_2.$ For simplicity of notation, when the second\nargument $R_2$ is fixed and there is no possibility of confusion, we\nshall set $\\varphi(R_1)= \\varphi(R_1, R_2)$ and $\\chi\n(R_1)=\\chi(R_1, R_2).$ By multiplying $I_1^*$ by the absolute\ninvariant $I_0\/ I_1$ and the relative invariant $S_1\/ R_0,$ we\nobtain a relative invariant of the form $F= \\varphi(S_1,\nS_0)\/\\varphi(R_0, S_0).$ By an application of Theorem ~\\ref{th:F0},\nit is easy to see that both $\\varphi(S_1)$ and $\\varphi(R_0)$ are\nrelative invariants, and they clearly have indices $m+r+1$ and\n$k+r+1,$ respectively. Moreover, $\\varphi (S_1)$ has the required\norder $\\mu+1,$ and it gives rise to the absolute invariant\n\\begin{equation}\\label{eq:chi}\n\\chi (S_1,\nS_0)=\\frac{\\left[\\varphi(S_1)\\right]^\\sigma}{S_0^{(m+r+1)}}.\n\\end{equation}\nOwing to the arbitrariness of the function $S_1,$ Eqs.\n ~\\eqref{eq:varphi} and ~\\eqref{eq:chi} show that starting with any\nrelative invariant $S_1$ of order $\\mu,$ we can construct an\nindefinite  sequence $\\varphi_q (S_1)= \\varphi^q (S_1)$ of relative\ninvariants and $\\chi_q(S_1)= \\chi^q (S_1)$ of absolute invariants,\neach having a general term of order $\\mu+q$. To write down the\nexpression for the general terms of these sequences, we only need to\nnote that $\\varphi_q (S_1)$ has index $\\theta (q)= m+ q (\\sigma+1).$\nConsequently, we have\n\\begin{align}\\label{eq:varphiseq}\n  \\varphi_q (S_1)&= \\theta (q-1) \\varphi^{q-1}(S_1) S_0' - r S_0\\, \\varphi^{q-1}(S_1)', \\qquad \\chi_q\n  (S_1)= \\frac{\\left[\\varphi_q(S_1)\\right]^\\sigma}{  S_0^{\\theta (q)}},\\\\\n\\intertext{ where } \\varphi^{q-1}(S_1)'&= d (  \\varphi^{q-1}(S_1))\/\ndx. \\notag\n\\end{align}\nA similar sequence of relative and absolute invariants can be\nobtained using the relative invariant $\\varphi(R_0, S_0)$. \\par\nIn the case of ODEs only one differential operator of the form\n$\\zeta D_x$ is required and once this operator is known, finding\nfundamental sets of invariants  only requires a suitable generating\nsystem of invariants, and these invariants must be explicitly\ncalculated. As already mentioned, earlier methods for finding these\ninvariants such as those used in ~\\cite{halph66} were very intuitive\nand quite informal, and they were designed only for linear\nequations. However, based on ideas suggested by Lie ~\\cite{lie1},\nand starting with some works undertaken by Ovsyannikov\n~\\cite{ovsy1}, the development of infinitesimal methods started to\ngrow and a number of Lie groups techniques for invariant functions\nhave been proposed in the recent scientific literature\n ~\\cite{ibra-nl, olvmf, olvgen, ndogftc}. These Lie groups methods\nprovide a systematic means for finding invariant functions and some\nof them have far reaching immediate applications (see ~\\cite{olvmf,\nndogftc}). For instance, the method of ~\\cite{ndogftc} that we shall\nuse for the explicit determination of the invariants also provides,\nin infinitesimal form,  the structure-preserving transformations of\nany differential equation. When it is applied to Eq.\n~\\eqref{eq:gnlin}, the coefficients $a_j(x)$ of the equation are\nconsidered as dependent variables on the same footing as $y$ and the\ninfinitesimal generators $X^0$ of the equivalence group $G_c$ takes\nthe form\n\\begin{equation}\\label{eq:infign}\nX^0 = f \\,\\partial_{x} + \\sum_{i=1}^n \\phi_i \\,\\partial_{a_i}.\n\\end{equation}\nThe explicit expression  of $X^0$ depends on various canonical forms\nadopted for the general linear ODE. By a change of variable of the\nform $x=z$ and $y= \\exp\\left(- \\int a_1 dz \\right) w,$ Eq.\n ~\\eqref{eq:gnlin} can be put in the form\n\\begin{equation}\\label{eq:nor}\nw^{(n)} + b_2 w^{(n-2)} + \\dots + b_n w=0\n\\end{equation}\nwhich is deprived of the term of second highest order, and in which\nthe $b_j = b_j(z)$ are the new coefficients. Similarly, by a change\nof variables of the form\n\\begin{subequations}\\label{eq:chg2schw}\n\\begin{align}\n\\set{z, x}&= \\frac{12}{n (n-1)(n+1)}  a_{2}, \\qquad y =\n\\exp\\left(- \\int a_1 d z\\right) w,\\\\\n\\intertext{ where } \\set{z, x} &= \\left(z\\,' z\\,^{(3)} - (3\/2)\nz\\,''^{\\,2} \\right)\\, z\\,'^{\\,-2}\n\\end{align}\n\\end{subequations}\nis the Schwarzian derivative, and where $z\\,' = d z\/ dx,$ we can put\nEq. ~\\eqref{eq:gnlin} into a form in which the terms of orders $n-1$\nand $n-2\\,$ do not appear. After the renaming of variables and\ncoefficients of the new equation using the same notation as for the\noriginal equation ~\\eqref{eq:gnlin}, the transformed equation takes\nthe form\n\\begin{equation}\\label{eq:schw}\ny^{(n)} + a_3 y^{(n-3)} + \\dots + a_n y=0.\n\\end{equation}\nIn fact, due to the prominence in size of invariants of differential\nequations and the rapid rate at which this size increases with the\nnumber of nonzero coefficients in the equation, it is customary to\nuse the canonical form ~\\eqref{eq:schw} of ~\\eqref{eq:gnlin} for the\ninvestigation of invariants ~\\cite{for-inv, brio}.\\par\nA determination of invariants of linear ODEs by means of the\ninfinitesimal generator of the form ~\\eqref{eq:infign} was made in\n ~\\cite{ndogschw} for equations of order up to five in the various\ncanonical forms ~\\eqref{eq:gnlin}, ~\\eqref{eq:nor}, and\n ~\\eqref{eq:schw}, but only for low orders of prolongation of the\noperator $X^0$ not exceeding three. We now undertake the explicit\ndetermination of a generating system of invariants of all orders by\nan application of formulas ~\\eqref{eq:varphi}-\\eqref{eq:varphiseq}.\nFor this purpose we shall adopt the canonical form ~\\eqref{eq:schw}\nfor which the invariants have a much simpler expression, and we\nrestrict our attention to the case of equations of order $n=5$ which\nhave the  form\n\\begin{equation}\\label{eq:sh5}\ny^{(5)} + a_3 y''+ a_4 y' + a_5 y=0.\n\\end{equation}\nFor every $n>3$, and for any order $p$ of prolongation of the\noperator $X^0,$ the number $\\Gamma$ of absolute invariants of Eq.\n\\eqref{eq:schw} can be shown ~\\cite{ndogschw} to be given by\n\\begin{equation}\\label{eq:nbschw}\n    \\Gamma = n+4 - p(n-2),\n\\end{equation}\nand this indicates that for a fixed value of $n,$ as the order of\nprolongation of $X^0$ increases by one unit, the number of absolute\ninvariants increases by $n-2,$ which is precisely the number of\ncoefficients in the equation. This means that a generating system of\nabsolute invariants should contain $n-2$ elements, which by virtue\nof ~\\eqref{eq:nbschw} can be taken to be functionally independent,\nprovided that none of the coefficients $a_j$ vanishes. For\n ~\\eqref{eq:sh5},  $X^0$ depends on three arbitrary constants $k_1,\nk_2$ and $k_3$ and has the explicit form\n\\begin{equation}\\label{eq:ifsh5}\n\\begin{split} X^0 &= \\left[ k_1+ x (k_2 + k_3 x)\\right] \\,\\partial_x - 3 a_3 (k_2 + 2 k_3 x)\\,\\partial_{a_3} \\\\\n & \\quad -2 \\left[-3 a_3 k_3 +2 a_4 (k_2 + 2 k_3 x)\\right]\\,\\partial_{a_4} + \\left[ -4 a_4 k_3\n - 5 a_5 (k_2+ 2 k_3 x)  \\right] \\,\\partial_{a_5}\\end{split}\n\\end{equation}\nand its first prolongation has four invariants given by\n\\begin{subequations}\\label{eq:ivp1sh5}\n\\begin{align}\nI_0 &=  \\frac{\\left(3a_5\\, a_3 - a_4^2  \\right)^3}{27 a_3^8} \\label{eq:schp0}\\\\\nI_1 &= \\frac{(- a_4 + a_3')^3}{a_3^4} \\\\\nI_2 &=  \\frac{\\left( 6 a_3 a_4' - a_4^2 - 6 a_4\\, a_3'\\right)^3}{216\\, a_3^8}\\\\\nI_3 &=\\frac{5 a_4^3 + 9 a_3^2\\, a_5' - 3 a_4\\, a_3(5 a_5  + 2 a_4')+\n3 a_4^2 \\, a_3'}{9 a_3^4}\n\\end{align}\n\\end{subequations}\nThe fundamental relative invariant of ~\\eqref{eq:schw} is readily\nseen to be given by $S_0=a_3(x),$ and Eq. ~\\eqref{eq:ivp1sh5} shows\nthat the four functions\n\\begin{align*}\nR_0 &= 3a_5\\, a_3 - a_4^2  \\\\\nS_1 &=- a_4 + a_3'  \\\\\nS_2 &= 6 a_3 a_4' - a_4^2 - 6 a_4\\, a_3' \\\\\nS_3 &= 5 a_4^3 + 9 a_3^2\\, a_5' - 3 a_4\\, a_3(5 a_5  + 2 a_4')+ 3\na_4^2 \\, a_3'\n\\end{align*}\nare relative invariants of respective index $8, 4, 8,$ and $12.$  In\nterms of these relative invariants, we have\n\\begin{align}\\label{eq:ifsh5sm}\nI_0 &=  \\frac{R_0^3 }{ S_0^8},\\qquad  I_1 = \\frac{S_1^3 }{ S_0^4},\n\\qquad I_2 = \\frac{S_2^3 }{ S_0^8},\\qquad  I_3 = \\frac{S_3}{ S_0^4}.\n\\end{align}\nMoreover, each of the four functions $R_0, S_1, S_2,$ and $S_3$ is\nof order at most one, and together they form a functionally\nindependent set. Consequently, on in accordance with\n ~\\eqref{eq:varphiseq}, $\\mathcal{B}= \\set{I_0, I_1, I_2, I_3}$ forms\na functionally independent generating system of absolute invariants\nwhich may be called a basis of absolute invariants of\n ~\\eqref{eq:schw}.\\par\n\n  It also follows that a fundamental set of absolute\ninvariants of order $p \\geq 1$ is given by the functions\n\\begin{equation}\\label{eq:ifsh5bs}\nI_0, \\quad \\chi_{k} (S_1),\\quad \\chi_{k} (S_2),\\quad \\chi_k\n(S_3),\\quad  \\text{ for $k=0, \\dots, p-1$}\n\\end{equation}\nand their number is $3 p +1,$ which is in accordance with equation\n ~\\eqref{eq:nbschw}. It follows from Eq. ~\\eqref{eq:varphiseq} that to\nfind explicit expressions for the absolute invariant $\\chi_k (S_j)$,\nwe only need to compute the corresponding relative invariant\n$\\varphi_k (S_j)$ and to determine its index. For $S_1,$  the\ncorresponding higher order relative invariants up to the order three\nare\n\\begin{align*}\n\\varphi (S_1) &=  4 (a_4 - a_3') a_3' + 3 a_3 (-a_4' + a_3'') \\\\\n\\varphi_2 (S_1)& = 32 (a_4 - a_3') a_3'^2 -\n  3 a_3 (9 a_4' a_3'+ (4 a_4 - 13 a_3') a_3'') \\\\\n       &\\quad  +\n  9 a_3^2 (a_4'' - a_3^{(3)})\n\\end{align*}\nand their indices are clearly $8$ and $12,$ respectively. The size\nof these two invariants are much more smaller compared to those\ncorresponding to $S_2$ and $S_3,$ simply because $S_1$ has a much\nsmaller size compared to $S_2$ and $S_3.$ For instance the\nexpression for $\\varphi_k (S_3)$ is about five times larger in size\nthan that for $\\varphi_k (S_1),$ for $k=1,2.$ It turns out that the\nexpression for the invariants computed explicitly by means of the\ninfinitesimal generator $X^0$ are considerably much smaller in size\nthan those computed from the indefinite sequence\n~\\eqref{eq:varphiseq}. Indeed, a direct computation shows that a\nfundamental set of differential invariants of order two is given by\n\\begin{align*}\nI_4 &=  \\frac{(7 a_4^2 - 14 a_4 a_3'+ 6 a_3 a_3'')^3}{216\\, a_3^8}\\\\\nI_5 &=  \\frac{4 a_4^3+ 24 a_4^2 a_3'+ 9 a_3^2 a_4''- 9 a_4 a_3 (3 a_4'+ a_3'')}{9\\, a_3^4}\\\\\nI_6 &=\\frac{(-18 a_4^4-18 a_4^3 a_3'+ 18 a_3^3 a_5''- 6 a_4 a_3^2\n(11 a_5' + 2 a_4'')+ a_4^2 a_3 (55 a_5+ 40 a_4' + 6 a_3''))^3}{5832\n \\, a_3^{16}}\n\\end{align*}\nand that for differential invariants of order three is given by\n\\begin{flushleft}\n\\begin{align*}\nI_7 &=\\frac{-14 a_4^3+ 42 a_4^2 a_3'- 36 a_4 a_3 a_3'' + 9 a_3^2 a_3^{(3)}}{9\\, a_3^4} \\\\\nI_8 &= \\frac{(-2 a_4^4-12 a_4^3 a_3'+ 3 a_4^2 a_3 (5 a_4'+ 3 a_3'')+\n2 a_3^3 a_4^{(3)}- 2 a_4 a_3^2(5 a_4'' + a_3^{(3)}))^3}{8\\, a_3^{16}} \\\\\nI_9 &= (S_{9,1}+ S_{9,2})^3\/(5832\\, a_3^{20})\n\\intertext{ where }\nS_{9,1} &= 35 a_4^5 + 45 a_4^4 a_3'- 10 a_4^3 a_3 (11 a_5+ 11 a_4'+\n3\na_3'') + 18 a_3^4 a_5^{(3)}\\\\\nS_{9,2}&= -12 a_4 a_3^3 (9 a_5''+ a_4^{(3)}+ 6 a_4^2 a_3^2 (33 a_5'\n+ 11 a_4''+ a_3^{(3)})^3).\n\\end{align*}\n\\end{flushleft}\nAs already mentioned  these absolute invariants of ~\\eqref{eq:sh5}\nare given in ~\\cite{ndogschw}, but only for the second prolongation\nof the operator $X^0,$ and not in a form in which the index can be\nreadily read off. They indeed appear to be smaller in size than\nthose derived from the recurrence relations ~\\eqref{eq:varphiseq},\nwhich indicates that a further simplification of the expression of\nthe function $\\varphi$ might be possible. However, such a\nsimplification might lead to more complicated recurrence equations\nfor the indefinite sequence of invariants. It's however naturally\nmuch easier to generate higher order invariants using the indefinite\nsequence ~\\eqref{eq:varphiseq}, rather than finding first the right\nprolongation of $X^0$ and then solving the corresponding system of\nequations to find the invariants. \\par\nSimilarly, using Eqs. ~\\eqref{eq:ifsh5sm}, ~\\eqref{eq:ivp1sh5}, and\n ~\\eqref{eq:varphiseq}, we can also determine, by invoking Corollary\n ~\\ref{co:4}, a fundamental set of relative invariants of all orders.\nFor instance,  fundamental relative invariants of order up to two of\n ~\\eqref{eq:sh5} are given by\n\\begin{subequations} \\label{eq:smp2sh5}\n\\begin{align}\n\\varphi_k (S_j, S_0),& \\quad j=1, 2, 3, \\quad k=0, 1\\\\\n\\varphi_k(R_0, S_0),&\\quad \\varphi_k(S_0, R_0), \\quad k=0, 1, 2.\n\\end{align}\n\\end{subequations}\nIncidently, Halphen ~\\cite{halph66} constructed by a different\nmethod an indefinite sequence of relative invariants similar to that\ngiven in ~\\eqref{eq:smp2sh5}, but for linear ODEs of order $4$ in\nthe canonical form ~\\eqref{eq:gnlin}. In particular the approach of\n ~\\cite{halph66} is not based on infinitesimal nor Lie groups\n methods and it aimed only\nat deriving and indefinite sequence of relative invariants from\nknown ones. Consequently, the sequence of relative invariants\nobtained in the said paper for equations of order $4$ is not\nassociated with the determination of a fundamental set of absolute\ninvariants.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe strong maximum principle of second order elliptic partial differential equations \nis due to Eberhard Hopf and it is one of the fundamental results in theory of differential equations. A very complete account of the developments in the area of maximum principles can be found in the works of Pucci and Serrin \\cite{PS04,PS07},\n where a thorough discussion and a complete bibliography is presented.\n\n\nIn this article we are interested in maximum principles for inequalities involving\ntwo operators $\\mathscr{L}_1$ and $\\mathscr{L}_0$ where\n$$\\mathscr{L}_1 u=\\Delta_\\infty u= \n\\sum_{i, j}\\partial_{x_i}u\\, \\partial_{x_i x_j} u\\, \\partial_{x_j}u\n\\quad \\text{and}\\quad \\mathscr{L}_0 u=\\frac{1}{|Du|^2}\\Delta_\\infty u.$$\n$\\mathscr{L}_1$ is popularly known as the {\\it infinity Laplacian} and $\\mathscr{L}_0$ is known\nas the {\\it normalized infinity Laplacian}. Though there are other variants of\ninfinity Laplacian operators one could consider, these two operators in particular, have received \nmore attention in the literature. Infinity Laplacian was first introduced in the pioneering works of G. Aronsson \\cite{AG1,AG2,AG3} and became quite popular in\nthe theory of partial differential equations. For more details on infinity Laplace \noperator we refer the readers to \\cite{ACJ,Lindqvist}.\nTo introduce our problem we consider a domain $\\mathcal{O}$ in $\\mathbb{R}^N$. Let $v$ be a\nnon-negative solution to\n\\begin{equation}\\label{E1.1}\n\\mathscr{L}_1v - K|Dv|^3 - f(v)\\leq 0 \\quad \\text{in}\\; \\mathcal{O},\n\\end{equation}\nor \n\\begin{equation}\\label{E1.2}\n\\mathscr{L}_0v - K|Dv| - f(v)\\leq 0 \\quad \\text{in}\\; \\mathcal{O},\n\\end{equation}\nwhere $K\\geq 0$ is a constant.\nBy a (sub or super) solution we always mean a viscosity (sub or super) solution (see Definition~\\ref{D2.1}).\nIn this article, $f:[0, \\infty)\\to [0, \\infty)$ is a given continuous, non-decreasing function and $f(0)=0$.\nLet $F(t)=\\int_0^t f(s)\\, \\mathrm{d}{s}$. A nonnegative solution $v$ is\nsaid to satisfy a strong maximum principle (SMP)\nin $\\mathcal{O}$ if $v(x_0)=0$ for some $x_0\\in \\mathcal{O}$ implies that $v\\equiv 0$ in $\\mathcal{O}$.\nWe establish the following SMP for \\eqref{E1.1} and \\eqref{E1.2}.\n\n\\begin{theorem}\\label{T1.1}\nConsider the following two conditions: for some (and thus for all) $\\delta>0$ we have\n\\begin{align}\n\\int_0^\\delta \\frac{1}{[F(s)]^{\\frac{1}{4}}}\\, \\mathrm{d}{s}\\,=\\,\\infty.\\label{EL1.1A}\n\\\\\n\\int_0^\\delta \\frac{1}{[F(s)]^{\\frac{1}{2}}}\\, \\mathrm{d}{s}\\,=\\,\\infty.\\label{EL1.1B}\n\\end{align}\nThen the following hold.\n\\begin{itemize}\n\\item[(a)] Assume \\eqref{EL1.1A} holds. If $v\\gneq 0$ is a solution of \\eqref{E1.1}, \nthen $v>0$ in $\\mathcal{O}$.\n\\item[(b)] Assume \\eqref{EL1.1B} holds. If $v\\gneq 0$ is a solution of \n\\eqref{E1.2},\nthen $v>0$ in $\\mathcal{O}$.\n\\end{itemize}\n\\end{theorem}\nTo compare the above result with the existing  results let us consider the\n$p$-Laplacian operator of the form\n\\begin{equation}\\label{p-lap}\n\\mathrm{div}(|Dv|^{p-2} Dv)- f(v)\\leq 0\\quad \\text{in}\\; \\mathcal{O}.\n\\end{equation}\nIt was proved by V\\'{a}zquez in \\cite{V84} that $v$ in \\eqref{p-lap} satisfies a \nSMP if for some $\\delta>0$ we have\n\\begin{equation}\\label{p-con}\n\\int_0^\\delta\\frac{1}{[F(s)]^{\\nicefrac{1}{p}}}\\, \\mathrm{d}{s}=\\infty.\n\\end{equation}\nIt turns out that this condition is also necessary for the validity of SMP; see Benilan-Brezis-Crandall \\cite{BBC} for $p=2$ and Diaz \\cite{D85} for all $p>1$. These results are then \nextended by Pucci, Serrin and Zou \\cite{PSZ} and\nby Pucci and Serrin \\cite{PS04} for operators of the form \n$$\\mathrm{div}(A(|Dv|) Dv)- f(v)\\leq 0\\quad \\text{in}\\; \\mathcal{O},$$\nfor a suitable continuous function $A$. For further developments in this direction \nwe refer to the works of Felmer-Montenegro-Quaas \\cite{FMQ}, Felmer-Quaas \\cite{FQ02}.\nOur Theorem~\\ref{T1.1} extends the SMP for infinity Laplacian operators.\n\n\nIt is shown in \\cite{PSZ} that when \\eqref{p-con} fails, that is, \n$$\\int_0^\\delta\\frac{1}{[F(s)]^{\\nicefrac{1}{p}}}\\, \\mathrm{d}{s}\\,<\\,\\infty, \n\\quad \\text{for some}\\; \\delta>0,$$\nthen a compact support principle (CSP) holds in the sense that any \nnonnegative solution $u$ of \n$$\\mathrm{div}(|Du|^{p-2} Du)- f(u)\\geq 0\\quad \\text{in}\\; B^c(0, r)$$\nwhich also vanishes at infinity, must vanish outside a compact set ( see also \\cite{FMQ,FQ02,PS04}). \nOur next result is about CSP which states that any nonnagtive\nsolution of \n$$\\mathscr{L}_i u + G(|Du|) - f(u)\\geq 0 \\quad \\text{in}\\; \\mathcal{O},$$\nthat vanishes at infinity, must have a compact support. Here $G:[0, \\infty)\\to [0, \\infty)$ is a continuous, nondecreasing function with $G(0)=0$.\nWe prove a stronger version of the CSP where we do not\nassume the solution to vanish at infinity.\n\\begin{theorem}\\label{T1.2}\nSuppose that $\\mathcal{O}$ is unbounded and $B^c(0, \\hat{r})\\subset \\mathcal{O}$ for some $\\hat{r}>0$. Let \n$f(s)>0$ for $s>0$. Then the following hold.\n\\begin{itemize}\n\\item[(a)] Define $\\Gamma(t)=\\int_0^{2t} G(s)\\mathrm{d}{s} + \\frac{1}{4} t^4$ and assume that\n\\begin{equation}\\label{ET2.2A0}\n\\int_0^1 \\frac{1}{\\Gamma^{-1}(F(s))}\\, \\mathrm{d}{s} < \\infty.\n\\end{equation}\nLet $u$ be a nonnegative, bounded function that solve \n\\begin{equation}\\label{EqnA}\n\\mathscr{L}_1 + G(|Du|) - f(u)\\geq 0 \\quad \\text{in}\\; \\mathcal{O}\\,.\n\\end{equation}\nThen there exists $R>0$ such that $u(x)=0$ for $|x|\\geq R$. \n\n\\item[(b)] Define $\\Gamma(t)=\\int_0^{2t} G(s)\\mathrm{d}{s} + \\frac{1}{2} t^2$ and assume that\n\\begin{equation}\\label{ET2.2B0}\n\\int_0^1 \\frac{1}{\\Gamma^{-1}(F(s))}\\, \\mathrm{d}{s} < \\infty.\n\\end{equation}\nLet $u$ be a nonnegative, bounded function that solve\n\\begin{equation*}\n\\mathscr{L}_0 + G(|Du|) - f(u)\\geq 0 \\quad \\text{in}\\; \\mathcal{O}.\n\\end{equation*}\nThen there exists $R>0$ such that $u(x)=0$ for $|x|\\geq R$. \n\\end{itemize}\n\\end{theorem}\n\nThe boundedness assumption in Theorem~\\ref{T1.2} can not be relaxed. For instance,\ntake $u(x)=e^{|x|}$, $G(s)=s^3$ and $f(s)=s^{3\\alpha}$ for $\\alpha\\in (0, 1)$. Then an\neasy calculation reveals  that for $x\\neq 0$\n\\begin{align*}\n\\Delta_\\infty u (x) + |Du(x)|^3 - f(u(x))= 2 e^{3|x|} - e^{3\\alpha|x|}>0.\n\\end{align*}\n\nNext we prove existence of a nonnegative solution with compact support. To compare\nit with Theorem~\\ref{T1.2} take $G(s)=Ks^3$ in Theorem~\\ref{T1.3}(a) and $G(s)=K s$ in (b) below.\n\n\\begin{theorem}\\label{T1.3}\nLet $\\mathcal{O}=B^c(0, 1)$ and \n$f(s)>0$ for $s>0$. Then the following hold.\n\\begin{itemize}\n\\item[(a)] Suppose that\n\\begin{equation}\\label{ET2.3A}\n\\int_0^1 \\frac{1}{(F(s))^{\\nicefrac{1}{4}}}\\, \\mathrm{d}{s} < \\infty.\n\\end{equation}\nThen for every $K>0$, there exists a $u\\gneq 0$ with compact support satisfying\n\\begin{equation}\\label{AB1}\n\\mathscr{L}_1u + K |Du|^3 - f(u)= 0 \\quad \\text{in}\\; \\mathcal{O}\\,.\n\\end{equation}\n\n\\item[(b)] Suppose that\n\\begin{equation}\\label{ET2.3B}\n\\int_0^1 \\frac{1}{(F(s))^{\\nicefrac{1}{2}}}\\, \\mathrm{d}{s} < \\infty.\n\\end{equation}\nThen for every $K>0$, there exists a $u\\gneq 0$ with compact support satisfying\n\\begin{equation}\\label{AB2}\n\\mathscr{L}_0u + K |Du| - f(u)= 0 \\quad \\text{in}\\; \\mathcal{O}\\,.\n\\end{equation}\n\\end{itemize}\n\\end{theorem}\n We note that the solution $u$ of \\eqref{AB1} (\\eqref{AB2}) also satisfies\n$$\\mathscr{L}_1u  - f(u)\\leq 0 \\quad \\text{in}\\; \\mathcal{O}, \n(\\mathscr{L}_0 u - f(u)\\leq  0 \\quad \\text{in}\\; \\mathcal{O}, \\text{respectively}).$$\nThus Theorem~\\ref{T1.3} also establishes the necessity of the conditions \\eqref{EL1.1A} and \\eqref{EL1.1B} in Theorem~\\ref{T1.1}.\n\nBefore conclude this section let us also mention the works \\cite{BD99,BB01,BCPR,SP} which\nalso consider maximum principles for infinity Laplacian operators. However, our maximum principles are quite different from the one studied in these works.\nOn the other hand, though infinite\nLaplacian can not be written in a divergence form, many ideas from \\cite{PSZ,FMQ}\nstill works for our model. The proofs of our results relies on two ingredients:\nthe ode method of \\cite{PSZ} and a new comparison theorem for infinity Laplacian\nrecently obtained by Biswas and Vo in \\cite{BV20,BV20a}.\n\n\\section{Proofs of Theorems~\\ref{T1.1},~\\ref{T1.2} and \\ref{T1.3}}\\label{S-proof}\nWe provide proofs of Theorems~\\ref{T1.1},~\\ref{T1.2} and \\ref{T1.3} in this section.\nWe begin with the definition of viscosity solution. Denote by\n$$\\widehat\\mathscr{L}_1 u=\\mathscr{L}_1 u + H(x, Du), \\quad \\text{and}\\quad\n\\widehat\\mathscr{L}_0 u =\\mathscr{L}_0 u + H(x, Du)\\,,$$\nwhere $H$ is a continuous function.\nAs mentioned before, in this article we deal with  viscosity solutions to the equations of the form\n\\begin{equation}\\label{E2.1}\n\\widehat\\mathscr{L}_i u + \\ell(x, u)\\,=\\, 0\\quad \\text{in}\\;  \\mathcal{O}, \\quad \\text{and}\n\\quad u=g\\quad \\text{on}\\; \\partial\\mathcal{O},\n\\end{equation}\nwhere $\\ell$ and $g$ are assumed  to be continuous and $i=1,2$. For a symmetric matrix $A$ we define\n$$M(A)=\\max_{\\abs{x}=1} \\langle x, A x\\rangle, \\quad m(A)= \\min_{\\abs{x}=1} \\langle x, A x\\rangle.$$\nThe open ball of radius $r$ centered at $z$ is denoted by $B(z, r)$. We use the notation $u\\prec_{z}\\varphi$ when $\\varphi$ touches $u$ from above exactly at the point $z$ i.e.,\nfor some open ball $B(z, r)$ around $z$ we have $u(y)<\\varphi(y)$ for $y\\in B(z, r)\\setminus\\{z\\}$ and\n$u(z)=\\varphi(z)$.\n\\begin{definition}[Viscosity solution]\\label{D2.1}\nAn upper-semicontinuous (lower-semicontinous) function $u$ in $\\bar{\\mathcal{O}}$ is said to be a viscosity sub-solution (super-solution) of \\eqref{E2.1}\n, written as $\\mathscr{L}_i u + \\ell(x, u)\\geq 0$ ($\\mathscr{L}_i u + \\ell(x, u)\\leq 0$),   if the followings are satisfied :\n\\begin{itemize}\n\\item[(i)] $u\\leq g$ on $\\partial \\mathcal{O}$ ($u\\geq g$ on $\\partial \\mathcal{O}$);\n\\item[(ii)] if $u\\prec_{x_0}\\varphi$ ($\\varphi\\prec_{x_0} u$ ) for\nsome point $x_0\\in\\mathcal{O}$ and a $\\mathcal{C}^2$ test function $\\varphi$, then \n\\begin{align*}\n& \\widehat\\mathscr{L}_i \\varphi(x_0) + \\ell(x_0, u(x_0))\\,\\geq\\, 0\\,,\n\\quad \\left(\\widehat\\mathscr{L}_i \\varphi(x_0) + \\ell(x_0, u(x_0))\\leq\\, 0,\\; resp., \\right);\n\\end{align*}\n\\item[(iii)] for $i=0$, if $u\\prec_{x_0}\\varphi$ ($\\varphi\\prec_{x_0} u$) and $D\\varphi(x_0)=0$ then \n\\begin{align*}\n& M(D^2\\varphi(x_0)) + H(x,  D \\varphi(x_0)) + \\ell(x_0, u(x_0))\\,\\geq\\, 0\\,,\n\\\\\n&\\left(m(D^2\\varphi(x_0)) + H(x,  D \\varphi(x_0)) + \\ell(x_0, u(x_0))\\leq\\, 0,\\; resp., \\right)\\,.\n\\end{align*}\n\\end{itemize}\nWe call $u$ a viscosity solution if it is both sub and super solution to \\eqref{E2.1}.\n\\end{definition}\nAs well known, one can replace the requirement of strict maximum (or minimum) above by non-strict maximum (or minimum). We would also require the notion of superjet and subjet from \\cite{CIL}. A second order \\textit{superjet} of $u$ at $x_0\\in\\mathcal{O}$ is defined as\n$$J^{2, +}_\\mathcal{O} u(x_0)=\\{(D\\varphi(x_0), D^2\\varphi(x_0))\\; :\\; \\varphi\\; \\text{is}\\; \\mathcal{C}^2\\; \\text{and}\\; u-\\varphi\\; \\text{has a maximum at}\\; x_0\\}.$$\nThe closure of a superjet is given by\n\\begin{align*}\n\\bar{J}^{2, +}_\\mathcal{O} u(x_0)&=\\Bigl\\{ (p, X)\\in\\mathbb{R}^N\\times\\mathbb{S}^{d\\times d}\\; :\\; \\exists \\; (p_n, X_n)\\in J^{2, +}_\\mathcal{O} u(x_n)\\; \\text{such that}\n\\\\\n&\\,\\qquad  (x_n, u(x_n), p_n, X_n) \\to (x_0, u(x_0), p, X)\\Bigr\\}.\n\\end{align*}\nSimilarly, we can also define closure of a subjet, denoted by $\\bar{J}^{2, -}_\\mathcal{O} u$. See \\cite{CIL} for more details.\n\nLet $H:[0, \\infty)\\to \\mathbb{R}$ be a continuous function. Denote by\n$\\mathcal{G}_i=\\mathscr{L}_i + H(|Du|)$. Our next ingredient is the following comparison principle which is a special case of \\cite[Theorem~2.1]{BV20}.\n\\begin{lemma}\\label{L2.1}\nLet $\\mathcal{O}$ be a bounded domain and $h, \\tilde{h}:\\mathcal{O}\\to \\mathbb{R}$ be \ncontinuous functions with $h>\\tilde{h}$ in $\\mathcal{O}$. Suppose that \n$\\mathcal{G}_i v - f(v)\\leq \\tilde{h}$ in $\\mathcal{O}$ \nand $\\mathcal{G}_i u - f(u)\\geq h$ in $\\mathcal{O}$. Then $v\\geq u$ on $\\partial \\mathcal{O}$\nimplies $v\\geq u$ in $\\bar{\\mathcal{O}}$.\n\\end{lemma}\n\n\\begin{proof}\nAs mentioned before, the proof follows from \\cite[Theorem~2.1]{BV20}. We just\nprovide a sketch of the proof here. Suppose, on the contrary, that \n$M=\\max_{\\bar{\\mathcal{O}}} (u-v)>0$. Now consider the coupling function\n$$w_\\varepsilon(x, y)= u(x)-v(y) -\\frac{1}{4\\varepsilon}|x-y|^4,\\quad\nx, y\\in\\bar\\mathcal{O}\\,.$$\nLet $M_\\varepsilon$ be the maximum of $w_\\varepsilon$ and \n$w_\\varepsilon(x_\\varepsilon, y_\\varepsilon)=M_\\varepsilon$.\nIt is then standard to show that (cf. \\cite[Lemma~3.1]{CIL})\n$$\\lim_{\\varepsilon\\to 0} M_\\varepsilon=M \\quad \\text{and}\\quad\n\\lim_{\\varepsilon\\to 0}\\frac{1}{4\\varepsilon} |x_\\varepsilon-y_\\varepsilon|^4=0.$$\nThus, without any loss of generality, we may assume that \n$x_\\varepsilon, y_\\varepsilon\\to z\\in\\bar\\mathcal{O}$ as $\\varepsilon\\to 0$. Otherwise, we may\nchoose a subsequence. Since $u-v\\leq 0$ on $\\partial\\mathcal{O}$ we must have $z\\in\\mathcal{O}$.\nDenote by \n$\\eta_\\varepsilon=\\frac{1}{\\varepsilon}|x_\\varepsilon-y_\\varepsilon|^2(x_\\varepsilon-y_\\varepsilon)$ and $\\theta_\\varepsilon(x, y)= \\frac{1}{4\\varepsilon}|x-y|^4$.\nIt then follows from \\cite[Theorem~3.2]{CIL} that for some $X, Y\\in\\mathbb{S}^{d\\times d}$ we have $(\\eta_\\varepsilon , X)\\in\\bar{J}^{2, +}_\\mathcal{O} u(x_\\varepsilon)$,\n$(\\eta_\\varepsilon, Y)\\in\\bar{J}^{2, -}_\\mathcal{O} v(y_\\varepsilon)$ and\n\\begin{equation}\\label{EL1.2A}\n\\begin{pmatrix}\nX & 0\\\\\n0 & -Y\n\\end{pmatrix}\n\\leq \nD^2\\theta_\\varepsilon(x_\\varepsilon, y_\\varepsilon) + \\varepsilon [D^2\\theta_\\varepsilon(x_\\varepsilon, y_\\varepsilon)]^2.\n\\end{equation}\nIn particular, we get $X\\leq Y$. Moreover, if $\\eta_\\varepsilon=0$, we have $x_\\varepsilon= y_\\varepsilon$. Then from \\eqref{EL1.2A} it follows that\n\\begin{equation}\\label{EL1.2B}\n\\begin{pmatrix}\nX & 0\\\\\n0 & -Y\n\\end{pmatrix}\n\\leq \n\\begin{pmatrix}\n0 & 0\\\\\n0 & 0\n\\end{pmatrix}.\n\\end{equation}\nNote that \\eqref{EL1.2B} implies that $X\\leq 0\\leq Y$ and therefore, $M(X)\\leq 0\\leq m(Y)$.\nApplying the definition of superjet and subjet\non $\\mathcal{G}_1$ we now obtain for $\\eta_\\varepsilon\\neq 0$\n\\begin{align}\\label{EL1.2C}\nh(x_\\varepsilon)&\\leq \\langle \\eta_\\varepsilon X, \\eta_\\varepsilon \\rangle\n + H(|\\eta_\\varepsilon|)-f(u(x_\\varepsilon))\\nonumber\n\\\\\n& \\leq  \\langle \\eta_\\varepsilon Y, \\eta_\\varepsilon \\rangle\n + H(|\\eta_\\varepsilon|) - f(u(x_\\varepsilon))\\nonumber\n\\\\\n&\\leq \\langle \\eta_\\varepsilon Y, \\eta_\\varepsilon \\rangle\n + H(|\\eta_\\varepsilon|) - f(v(y_\\varepsilon))\\nonumber\n\\\\\n&\\leq \\tilde{h}(y_\\varepsilon),\n\\end{align}\nwhere in the third line we use the fact $f(v(y_\\varepsilon))\\leq f(u(x_\\varepsilon))$.\nLetting $\\varepsilon\\to 0$ we obtain $h(z)\\leq \\tilde{h}(z)$ which is a contradiction \nto our hypothesis.\nSimilar argument also works  for $\\mathcal{G}_0$. This completes the proof.\n\\end{proof}\n\nFollowing lemma is a key ingredient in the proof of\nTheorem~\\ref{T1.1}.\n\\begin{lemma}\\label{L1.1}\nFor every $\\varepsilon, K>0$ and $R\\in (0, 1)$ we have $\\alpha<0$ so that\n\\begin{itemize}\n\\item[(i)] under \\eqref{EL1.1A}, there is a twice continuously differentiable solution\n$\\varphi$ satisfying\n\\begin{align}\n((\\varphi^\\prime)^3)^\\prime + K (\\varphi^\\prime)^3 - f(\\varphi)+\\alpha&=0\\quad\n\\text{in}\\; ({R}\/{2}, R+\\varepsilon_1),\\quad\n\\varphi^\\prime(R)=\\alpha, \\quad \\varphi(R)=0\\,,\\label{EL1.1C}\n\\\\\n&0<\\varphi<\\varepsilon, \\quad \\varphi^\\prime<0\\quad \\text{in} \\; ({R}\/{2}, R)\\,,\\label{EL1.1D}\n\\end{align}\nfor some $\\varepsilon_1>0$.\n\n\\item[(i)] under \\eqref{EL1.1B}, there is a twice continuously differentiable solution\n$\\varphi$ satisfying\n\\begin{align}\n\\varphi^{\\prime\\prime} + K \\varphi^\\prime - f(\\varphi)+\\alpha &=0\\quad\n\\text{in}\\; ({R}\/{2}, R+\\varepsilon_1),\\quad\n\\varphi^\\prime(R)=\\alpha, \\quad \\varphi(R)=0\\,,\\label{EL1.1E}\n\\\\\n&0<\\varphi<\\varepsilon, \\quad \\varphi^\\prime<0\\quad \\text{in} \\; ({R}\/{2}, R)\\,,\\label{EL1.1F}\n\\end{align}\nfor some $\\varepsilon_1>0$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nWe only find $\\varphi$ satisfying \\eqref{EL1.1C}-\\eqref{EL1.1D} and the proof for\n\\eqref{EL1.1E}-\\eqref{EL1.1F} would be analogous. The\nproof of \\eqref{EL1.1C}-\\eqref{EL1.1D} actually follows from the argument of \\cite[Lemma~2]{PSZ}. Nevertheless, we provide a proof to keep the article self-contained.\nDenote by $f_\\alpha = f-\\alpha$  for $\\alpha<0$. Also, we extend the domain $f_\\alpha$ \nto $\\mathbb{R}$ by setting $f_\\alpha(x)=-\\alpha$ for $x<0$.\nFirst we note that existence of a local solution of \\eqref{EL1.1C} follows from \nthe Schauder-Tychonoff fixed point theorem. In fact, for any $(\\xi, \\gamma)\\in \\mathbb{R}\\times\\mathbb{R}$,\nconsider the map $T:\\mathcal{C}[t_0-\\beta, t_0] \\to \\mathcal{C}[t_0-\\beta, t_0]$ defined as\n\\begin{equation}\\label{EL1.1G}\n(Tg)(t) = \\xi - \\int_t^{t_0} \\left(e^{K(t_0-s)}\\gamma^3 - \n\\int_{s}^{t_0} e^{K(\\zeta-s)} f_\\alpha(g(\\zeta))\\, \\mathrm{d}\\zeta\\right)^{\\nicefrac{1}{3}} \\mathrm{d}{s}\\,.\n\\end{equation}\nNow given positive $M_1, M_2$ we can find $\\beta>0$ so that\nfor any $|\\xi|\\leq M_1$, $|\\gamma|\\leq M_2$, $T$ satisfies\nthe condition of Schauder-Tychonoff fixed point theorem and hence, it has a fixed point. Set $\\xi=0, \\gamma=\\alpha$ and find a fixed point $\\varphi$ of $T$ in $[R-\\beta\/2, R+\\beta\/2]$.\nNext, setting $t_0=R-\\beta\/2, \\xi=\\varphi(t_0), \\gamma=\\varphi'(t_0)$, we can\nextend $\\varphi$ to $(R-\\beta, R+\\beta\/2)$ provided $|\\varphi(t_0)|\\leq M_1$\nand $|\\varphi'(t_0)|\\leq M_2$. Let $(R_0, R+\\beta\/2)$ be the maximal interval on\nwhich $\\varphi$ can be defined by repeating the above scheme. It is evident that \n$\\varphi$ is continuously differentiable and \n$$\\varphi'(t) = \\left(e^{K(R-t)}\\alpha^3 - \n\\int_{t}^{R} e^{K(\\zeta-s)} f_\\alpha(\\varphi(\\zeta))\\, \\mathrm{d}\\zeta\\right)^{\\nicefrac{1}{3}}<0.\n$$\nThus $\\varphi$ is strictly deceasing and $\\varphi^\\prime< 0$ in $(R_0, R+\\beta\/2)$.\nIt is then easily seen from \\eqref{EL1.1G} that $\\varphi$ is twice continuously differentiable and satisfies \\eqref{EL1.1C} in $(R_0, R+\\beta\/2)$. Thus $\\varphi$\nsatisfies \\eqref{EL1.1C} in $(R_0, R+\\beta\/2)$. Let $R_1\\in(R, R_0]$ be the maximal\nnumber so that $\\varphi$ satisfies \\eqref{EL1.1C}-\\eqref{EL1.1D} in $(R_1, R+\\beta\/2)$.\nTo complete the proof we only need show that if we choose $|\\alpha|$ small enough then we can have $R_1\\leq R\/2$. Suppose, on the contrary, that $R_1> R\/2$. Given the maximality of $R_1$, one of the following to possibilities hold:\n\\begin{equation}\\label{EL1.1H}\n\\mathrm{(a)}\\, \\lim_{t\\to R_1+} \\varphi(t)=\\varepsilon,\n\\quad \\mathrm{(b)}\\, \\lim_{t\\to R_1+} |\\varphi^\\prime(t)|> M_2.\n\\end{equation}\nIt is easily seen from \\eqref{EL1.1C} that $\\varphi^{\\prime\\prime}>0$ in $(R_1, R)$ and\nthus $\\varphi^\\prime$ is increasing.\nLetting \n$$F_\\alpha(t) =\\int_0^t f_\\alpha(s) \\mathrm{d}{s},$$\nand multiplying \\eqref{EL1.1C} by $\\varphi^\\prime$ we see that\n$$\\left(e^{\\tilde{K} t} (\\varphi^\\prime)^4\\right)^\\prime - \\frac{4}{3}e^{\\tilde{K} t}(F_\\alpha(\\varphi))^\\prime=0,$$\nwhere $\\tilde{K}=\\frac{4}{3}K$. Since \n$(F_\\alpha(\\varphi))^\\prime=f_\\alpha(\\varphi) \\varphi^\\prime<0$, we get \n$$e^{\\tilde{K} R} \\alpha^4- e^{\\tilde{K} t} (\\varphi^\\prime(t))^4 +\n\\frac{4}{3}e^{\\tilde{K} R}F_\\alpha(\\varphi(t))\\geq 0\\,.$$\nThis of course, gives us\n\\begin{equation}\\label{EL1.1I}\n(\\varphi^\\prime(t))^4\\leq e^{\\frac{\\tilde{K}R}{2}} \\alpha^4 + \ne^{\\frac{\\tilde{K}R}{2}}\\frac{4}{3} F_\\alpha(\\varphi(t)), \\quad t\\in (R_1, R)\\,.\n\\end{equation}\nNow, without any loss of generality, we may take $\\varepsilon\\in (0, 1)$. \nTherefore, at the beginning, if we choose $M_2$ large enough to satisfy\n$$ \\left[e^{\\frac{\\tilde{K}R}{2}} \\alpha^4 + \n\\frac{4}{3} e^{\\frac{\\tilde{K}R}{2}} \\max_{s\\in [0, 1]}F_\\alpha(s)\\right]^{\\nicefrac{1}{4}}< M_2, $$\n possibility (b) in \n\\eqref{EL1.1H} can not occur before (a). In other words, if we have $R_1>R\/2$, then \n(a) is the only possibility. Thus it is enough to consider (a).\nRestrict $\\varepsilon<\\delta$ where $\\delta$ is given by\n\\eqref{EL1.1A}. We can further restrict $\\varepsilon$ to satisfy\n$F_\\alpha(\\varepsilon)<\\delta$. \nChoose $\\alpha$ small enough so that \n$\\widehat{\\varepsilon}= F_\\alpha^{-1}(\\alpha^4)<\\varepsilon$. Then we can find\n$\\hat{R}\\in (R_1, R)$ satisfying $\\varphi(\\hat{R})=\\hat{\\varepsilon}$ and\n$\\varphi(t)\\geq \\hat\\varepsilon$ in $(R_1, \\hat{R})$. Then\n$$F_\\alpha(\\varphi(t))\\geq F_\\alpha (\\hat\\varepsilon)=\\alpha^4\\quad \\text{in}\n\\; (R_1, \\hat{R}).$$\nUsing \\eqref{EL1.1I} we then obtain\n$$ -\\varphi^\\prime(t) \\leq \\left(\\frac{7}{3}\\right)^\\frac{1}{4}\\,\ne^{\\frac{\\tilde{K} R}{8}} [F_\\alpha(\\varphi(t))]^\\frac{1}{4}\n\\quad \\text{in}\\; (R_1, \\hat{R}).$$\nIntegrating both sides we have\n$$\n\\int^{\\hat{R}}_{R_1} \\frac{-\\varphi^\\prime(t)}{[F_\\alpha(\\varphi(t))]^\\frac{1}{4}}\\, \\mathrm{d}{t}\n\\leq \\, \\left(\\frac{7}{3}\\right)^\\frac{1}{4}\\,\ne^{\\frac{\\tilde{K} R}{8}} \\frac{R}{2},$$\nwhich in turn, gives\n\\begin{equation}\\label{AB3}\n\\int^{\\varepsilon}_{\\hat\\varepsilon} \\frac{1}{[F(t)-\\alpha t]^\\frac{1}{4}}\\, \\mathrm{d}{t}=\n\\int^{\\varepsilon}_{\\hat\\varepsilon} \\frac{1}{[F_\\alpha(t)]^\\frac{1}{4}}\\, \\mathrm{d}{t}\n\\leq \\, \\left(\\frac{7}{3}\\right)^\\frac{1}{4}\\,\ne^{\\frac{\\tilde{K} R}{8}} \\frac{R}{2}.\n\\end{equation}\nSince $\\alpha\\to 0$ implies $\\hat{\\varepsilon}\\to 0$, using monotone convergence \ntheorem we note that\n$$\\int^{\\varepsilon}_{\\hat\\varepsilon} \\frac{1}{[F(t)-\\alpha t]^\\frac{1}{4}}\\, \\mathrm{d}{t}\n\\to \\int^{\\varepsilon}_{0} \\frac{1}{[F(t)]^\\frac{1}{4}}\\, \\mathrm{d}{t}, \\quad \\text{as}\\; \\alpha\\to 0.$$\nBut the limit is $\\infty$ by \\eqref{EL1.1A}. This is a contradiction to \\eqref{AB3}.\nHence (a) in \\eqref{EL1.1H} is also not possible for small enough $\\alpha$.\nThus $R_1\\leq R\/2$ which completes the proof.\n\\end{proof}\n\nNow we are ready to prove our main results. We start with the proof of Theorem~\\ref{T1.1}.\n\\begin{proof}[{\\bf Proof of Theorem~\\ref{T1.1}}]\nWe only provide a proof for (a) and the proof for (b) would be analogous.\nSuppose, on the contrary, that the set $\\{x\\in\\mathcal{O}\\; :\\; v(x)=0\\}$ is non-empty. Then\nsince $v\\neq 0$, we can find a ball $B(x_0, R)\\subset \\mathcal{O}$ such that\n$v>0$ in $B(x_0, R)$ and \n$\\overline{B(x_0, R)}\\cap\\{x\\in\\mathcal{O}\\; :\\; v(x)=0\\}\\neq\\emptyset$. Without loss of generality, assume that $R\\in (0, 1)$. Choose $\\varepsilon< \\min_{\\overline{B}(x_0, R\/2)} v$.\nUsing Lemma~\\ref{L1.1} we now find a twice continuously differentiable function\n$\\varphi$ satisfying\n\\begin{align}\n(\\varphi^\\prime)^2\\varphi^{\\prime\\prime} + K (\\varphi^\\prime)^3 - f(\\varphi)+\\alpha&=0\\quad\n\\text{in}\\; ({R}\/{2}, R+\\varepsilon_1),\\quad\n\\varphi^\\prime(R)=\\alpha, \\quad \\varphi(R)=0\\,,\\label{ET2.1C}\n\\\\\n&0<\\varphi<\\varepsilon, \\quad \\varphi^\\prime<0\\quad \\text{in} \\; ({R}\/{2}, R)\\,,\\label{ET2.1D}\n\\end{align}\nfor some $\\varepsilon_1>0$ and $\\alpha<0$. Let $u(x)= \\varphi(|x-x_0|)$. Then in $B^c(x_0, R\/2)$\nwe have\n\\begin{align*}\nDu(x)&= \\frac{x-x_0}{|x-x_0|}\\varphi^\\prime(|x-x_0|)\n\\\\\n\\partial_{x_ix_j}u &= \\frac{(x-x_0)_i(x-x_0)_j}{|x-x_0|^2}\\varphi^{\\prime\\prime}(|x-x_0|)+ \\varphi^\\prime(|x-x_0|)\\left(\\frac{\\delta_{ij}}{|x-x_0|}-\\frac{(x-x_0)_i(x-x_0)_j}{|x-x_0|^3}\\right).\n\\end{align*}\nUsing \\eqref{ET2.1C}-\\eqref{ET2.1D} we then have\n\\begin{align}\\label{ET2.1E}\n\\mathscr{L}_1 u - K |Du|^3 - f(u)\n& =\n(\\varphi^\\prime)^2(|x-x_0|)\\varphi^{\\prime\\prime}(|x-x_0|)\n- K |\\varphi^{\\prime}|^3 - f(\\varphi)\\nonumber\n\\\\\n&= (\\varphi^\\prime)^2(|x-x_0|)\\varphi^{\\prime\\prime}(|x-x_0|)\n+ K (\\varphi^{\\prime})^3 - f(\\varphi)=-\\alpha>0.\n\\end{align}\nUsing Lemma~\\ref{L2.1} we then have $u\\leq v$ for $R\/2\\leq|x-x_0|\\leq R$. Also,\nnote that for $R\\leq |x-x_0|\\leq R+\\varepsilon_1$, $u(x)\\leq 0$. Thus, $u$ touches\n$v$ from below at some point, say $z$, on the sphere $|x-x_0|=R$. Applying the definition of viscosity solution we must have\n$$\\mathscr{L}_1 u(z) - K |Du(z)|^3 - f(u(z))=\\mathscr{L}_1 u(z) - K |Du(z)| - f(v(z))\\leq 0,$$\nwhich contradicts \\eqref{ET2.1E}. Thus $\\{x\\in\\mathcal{O}\\; :\\; v(x)=0\\}=\\emptyset$, completing\nthe proof.\n\\end{proof}\n\n\nNext we prove Theorem~\\ref{T1.2}.\n\n\\begin{proof}[{\\bf Proof of Theorem~\\ref{T1.2}}]\nFirst we consider (a). We start by assuming that $\\lim_{|x|\\to\\infty} u(x)=0$ and\nestablish a compact support principle.\nFor this proof we borrow the ideas from \\cite{FMQ}. The main idea is to construct\na nonnegative super-solution to \\eqref{EqnA} with a compact support. To do so we again use the ODE technique.\nNote that by monotonicity of $f$ we have $F(a t)\\leq a F(t)$ for every $a\\in [0, 1]$, \nsince\n$$F(a t)=\\int_0^{at} f(s) \\mathrm{d}{s}=a\\int_0^t f(as) \\mathrm{d}{s}\\leq a\\int_0^t f(s) \\mathrm{d}{s}=aF(t).$$\nThus, by \\eqref{ET2.2A0}, we have\n$$\\int_0^1 \\frac{1}{\\Gamma^{-1}(4^{-1}F(s))}\\, \\mathrm{d}{s}<\\infty.$$\nDefine a continuous function $\\varphi$ by\n$$t=\\int_0^{\\varphi(t)} \\frac{1}{\\Gamma^{-1}(4^{-1}F(s))}\\, \\mathrm{d}{s}\\,. $$\nNote that $\\varphi$ is strictly increasing with $\\varphi(0)=0$. Also,\n\\begin{equation}\\label{ET2.2C0}\n1=\\frac{\\varphi'(t)}{\\Gamma^{-1}(4^{-1}F(\\varphi(t)))}\n\\quad \\Rightarrow\\quad \\Gamma(\\varphi'(t))=\\frac{1}{4} F(\\varphi(t)).\n\\end{equation}\nSince $\\Gamma, F, \\varphi$ are strictly increasing, we have $\\varphi'$ strictly increasing and $\\varphi'(0)=0$. Hence for $t\\leq 1$ we have\n$\\varphi(t)=\\int_0^t \\varphi'(s)\\, \\mathrm{d}{s}\\leq \\varphi'(t)$.\nIt is also evident from \\eqref{ET2.2C0} that \n$\\varphi'$ is continuously differentiable for $t>0$. Therefore, using \\eqref{ET2.2C0}\nand the fact $G$ is nondecreasing, we obtain for $t\\in [0, 1]$ that\n\\begin{align*}\nG(\\varphi'(t))\\varphi'(t)\\leq \\int_{\\varphi'(t)}^{2\\varphi'(t)} G(\\varphi'(s))\\, \\mathrm{d}{s}\n&\\leq \\Gamma(\\varphi'(t))=\\frac{1}{4} F(\\varphi(t))\\leq \\frac{1}{4} f(\\varphi(t))\\varphi(t)\\leq \\frac{1}{4} f(\\varphi(t))\\varphi'(t),\n\\end{align*}\nwhich in turn, implies\n\\begin{equation}\\label{ET2.2D0}\nG(\\varphi')\\leq \\frac{1}{4} f(\\varphi(t))\\quad \\text{for all $t>0$ small}.\n\\end{equation}\nSince $\\varphi^{\\prime\\prime}\\geq 0$ for $t>0$, differentiating \\eqref{ET2.2C0} we have\n$$(\\varphi'(t))^3\\varphi^{\\prime\\prime}(t) \n\\leq (\\Gamma(\\varphi(t))'=\\frac{1}{4} f(\\varphi(t))\\varphi'(t),$$\ngiving us\n\\begin{equation}\\label{ET2.2E0}\n(\\varphi'(t))^2\\varphi^{\\prime\\prime}(t)\\leq \\frac{1}{4} f(\\varphi(t))\n\\quad \\text{for all $t>0$ small}.\n\\end{equation}\nCombining \\eqref{ET2.2D0}-\\eqref{ET2.2E0} we find $r_\\circ>0$ such that \n\\begin{equation}\\label{ET2.2F0}\n(\\varphi'(t))^2\\varphi^{\\prime\\prime}(t) + G(\\varphi'(t))-2^{-1}f(\\varphi(t))\n\\leq 0 \\quad \\text{for all}\\; t\\in (0, r_\\circ),\n\\end{equation}\nand $\\varphi(0)=\\varphi'(0)=0$. We extend $\\varphi$ on $(-\\infty, 0]$ by setting\n$\\varphi(t)=0$ for $t\\leq 0$. It is easily seen that $\\varphi$ is continuously\ndifferentiable in $(-\\infty, r_\\circ)$. Now for any $R>0$, we let \n$v(x)=\\varphi(R+r_\\circ-|x|)$ for $|x|\\geq R$. Using \\eqref{ET2.2F0} and the calculations in \\eqref{ET2.1E}\nwe see that for $ R<|x|< R+r_\\circ$ we have\n\\begin{align}\\label{ET2.2G0}\n\\mathscr{L}_1 v + G(|Dv|) - 2^{-1}f(v)\n =\n(\\varphi^\\prime)^2(R+r_\\circ-|x|)\\varphi^{\\prime\\prime}(R+r_\\circ-|x|)\n+ G(\\varphi^{\\prime}) - 2^{-1}f(\\varphi)\n\\leq 0.\n\\end{align}\nWe claim that \n\\begin{equation}\\label{ET2.2H0}\n\\mathscr{L}_1 v + G(|Dv|) - 2^{-1}f(v)\\leq 0 \\quad \\text{for}\\; |x|> R\\,,\n\\end{equation}\nin viscosity sense. When $ R<|x|< R+r_\\circ$, \\eqref{ET2.2H0} follows\nfrom \\eqref{ET2.2G0}. Again, for $|x|> R+r_\\circ$, \\eqref{ET2.2H0} is evident  as\n$v$ is identically zero there.\nSo we consider the case where $|x|= R+r_\\circ$. Let $\\chi$ be a $\\mathcal{C}^2$ test with\n$\\chi\\prec_{x} v$. Since $v$ is $\\mathcal{C}^1$ we have $D\\chi(x)=Dv(x)=0$. Hence\n$$\\mathscr{L}_1\\chi(x) + G (|D\\chi(x)|)- 2^{-1}f(v(x))= 0,$$\nimplying $v$ is supersolution.  This \ngives us \\eqref{ET2.2H0}.\n\nNow we complete the proof. Let $\\beta=\\varphi(r_\\circ)>0$. Since $u(|x|)\\to 0$\nas $|x|\\to\\infty$, we find $R$ so that $u(x)<\\beta $ for $|x|\\geq R$.\nDefine \n$v_\\epsilon(x)=v(x) + \\epsilon$. Since $f$ is non-decreasing, we get from \\eqref{ET2.2H0} that\n$$\\mathscr{L}_1 v_\\epsilon + G (|Dv_\\epsilon|)-f(v_\\epsilon) \n\\leq 2^{-1}f(v)-f(v_\\epsilon)\\leq -2^{-1}f(v_\\epsilon)\\,,\n\\quad |x|> R.$$\nNow we choose $R_\\epsilon>R+r_\\circ$ large enough so that $u(x)<\\epsilon$\nfor $|x|\\geq R_\\epsilon$. Applying Lemma~\\ref{L2.1} in $\\{R<|x|<\nR_\\epsilon\\}$ we obtain $u\\leq v_\\epsilon=v+\\epsilon$ for $|x|\\geq R$.\nNow let $\\epsilon\\to 0$ to conclude that $u\\leq v$ for $|x|\\geq R$\nwhich implies $u(x)=0$ for $|x|\\geq R+r_\\circ$. This completes the proof of\n(a) under the assumption $\\lim_{|x|\\to\\infty} u(x)=0$.\n\nNow consider a bounded solution $u$ to\n\\eqref{EqnA} and we show that $\\lim_{|x|\\to\\infty} u(x)=0$. Then the conclusion \nof (a) follows from the first part of the proof.\nSuppose, on the contrary, that $\\limsup_{|x|\\to\\infty} u(x)=M>0$ and $f(M)=3\\kappa$.\nGiven $x_0\\in\\mathbb{R}^N$, we define $\\xi(x)=r^{-2}|x-x_0|^2-1$. Then \n$\\xi<0$ in $B(x_0, r)$ and vanishes at the boundary $\\partial B(x_0, r)$. Also,\n$$\\mathscr{L}_1\\xi + G(|D\\xi|)= 8 r^{-6}|x-x_0|^2 + G(2r^{-2}|x-x_0|)<\\kappa\n\\quad \\text{in}\\; B(x_0, r),$$\nprovided $r$ is large. Now choose a point $x_0\\in\\mathcal{O}$ such that $B(x_0, r)\\subset\\mathcal{O}$\nand $u(x_0)> M-\\epsilon$ where $\\epsilon\\in (0, 1\/2)$ is small enough to satisfy \n$f(M-\\epsilon)>2\\kappa$.  We may also assume that $\\sup_{\\overline{B(x_0, r)}} u< M+ 1\/2$.\n Now we translate $\\xi$ so that it touches $u$ in $B(x_0, r)$ from above.\nTo do so, we define\n$$\\beta=\\inf\\{\\gamma\\in[M-2\\epsilon, M+2]\\; :\\; \\gamma+\\xi > u\\; \\; \\text{in}\\; B(x_0, r)\\}.$$\nClearly, $\\beta\\geq M+1-\\epsilon$ since $M+1-\\epsilon+\\xi(x_0)=M-\\epsilon< u(x_0)$.\nHence $v(x):=\\beta+\\xi(x)=\\beta\\geq M+1-\\epsilon> u(x)$ on $\\partial B(x_0, r)$.\n$u$ being upper-semicontinuous, $v$ must touch $u$ inside $B(x_0, r)$, say at a point $z\\in B(x_0, r)$.\nThen applying the definition of viscosity solution we obtain\n$$f(v(z))=f(u(z))\\leq \\mathscr{L}_1 v + G(|Dv|)\\leq \\kappa.$$\nSince $v(z)\\geq M+1-\\epsilon+\\xi(z)\\geq M-\\epsilon$, we get from above that $\\kappa\\geq f(v(z))\\geq f(M-\\epsilon)>2\\kappa$ which is a contradiction. Thus we must have $\\lim_{|x|\\to\\infty} u(x)=0$. Hence the proof.\n\n\n\n\nNext we consider (b). The proof is similar to (a). Following a similar argument \nof \\eqref{ET2.2F0} we would obtain\n\\begin{equation*}\n\\varphi^{\\prime\\prime}(t) + G(\\varphi'(t))-2^{-1}f(\\varphi(t))\n\\leq 0 \\quad \\text{for}\\; t\\in (0, r_\\circ)\\,.\n\\end{equation*}\nIt can be easily seen from this equation that \n$\\lim_{t\\to 0+}\\varphi^{\\prime\\prime}(t)=\\varphi^{\\prime\\prime}(0)=0$. Thus the extension of $\\varphi$ is twice continuously differentiable in $(-\\infty, r_\\circ)$.\nNow we can follow the arguments of (a) to complete the proof.\n\\end{proof}\n\n\nFinally, we prove Theorem~\\ref{T1.3}.\n\\begin{proof}[{\\bf Proof of Theorem~\\ref{T1.3}}]\nAs before, we only prove (a) and the proof for (b) would be analogous.\nLet \n$$R= \\int_0^1 \\frac{1}{[H(s)]^{\\nicefrac{1}{4}}}\\, \\mathrm{d}{s},$$\nwhere $H(t)=\\int_0^t h(s) \\mathrm{d}{s}$ and $h(s)=4\\kappa f(s)$ for some $\\kappa>0$ to \nbe chosen later.\nWe define a continuous function $\\varphi:[0, R]\\to [0, \\infty)$ by\n$$r=\\int_{\\varphi(r)}^1 \\frac{1}{[H(s)]^{\\nicefrac{1}{4}}}\\, \\mathrm{d}{s}.$$\n It is evident that $\\varphi$ takes values in $[0, 1]$ and is strictly decreasing.\nDifferentiating  we obtain\n$$\\frac{-\\varphi'(r)}{[H(\\varphi(r))]^{\\nicefrac{1}{4}}}=1 \\quad \\text{for}\\; 0<r< R.$$\nSince $\\varphi'\\neq 0$ in $(0, R)$, differentiating once again we get\n\\begin{equation}\\label{ET2.2A}\n(\\varphi')^2 \\varphi^{\\prime\\prime} - \\frac{1}{4}h(\\varphi(r))=0\\quad \\text{in}\\; (0, R).\n\\end{equation}\nSince $\\varphi(R)=\\varphi^\\prime(R)=0$, from \\eqref{ET2.2A} we have\n\\begin{equation}\\label{ET2.2B}\n\\varphi(r)=\\int_r^R \\left[\\int_t^R \\frac{3}{4} h(\\varphi(s)) \\mathrm{d}{s}\\right]^{\\nicefrac{1}{3}}\\, \\mathrm{d}{t}\\quad r\\in (0, R]\\,.\n\\end{equation}\nChoose $\\delta>0$ small enough so that $8e^{-3K\\delta}\\geq 1$, where $K$ is same as in Theorem~\\ref{T1.3}. Now consider a map $T:\\mathcal{C}[R-\\delta, R]\\to \\mathcal{C}[R-\\delta, R]$\ngiven by\n\\begin{equation}\\label{ET2.2C}\n(Tg)(t) = \\int_t^{R} \\left[\\int_{s}^{R} 6 e^{3K(s-\\zeta)} h(g(\\zeta))\\, \\mathrm{d}\\zeta\\right]^{\\nicefrac{1}{3}} \\mathrm{d}{s}\\,.\n\\end{equation}\nIt is easily seen that $T$ is a continuous function. Also, if $g\\geq \\varphi$, then\nsince $h$ is non-decreasing using \\eqref{ET2.2B} we get\n$$(Tg)(t)\\geq \\int_t^{R} \\left[\\int_{s}^{R} 6 e^{-3K\\delta} h(\\varphi(\\zeta))\\, \\mathrm{d}\\zeta\\right]^{\\nicefrac{1}{3}} \\mathrm{d}{s}\n\\geq \\int_t^{R} \\left[\\int_{s}^{R} 3 h(\\varphi(\\zeta))\\, \\mathrm{d}\\zeta\\right]^{\\nicefrac{1}{3}} \\mathrm{d}{s}=\\varphi(t)\\,.$$\nDenote by $M=\\sup_{s\\in [0, 1]} h(s)$. Then, restricting $\\delta$ small enough\nwe see that if $\\sup_{t\\in [R-\\delta, R]}|g(t)|\\leq 1$ then\n\\begin{align*}\n|(Tg)(t)|\\leq (6M)^{\\frac{1}{3}}\\int_t^{R} (R-s )^{\\nicefrac{1}{3}} \\mathrm{d}{s}\n= \\frac{3}{4}(6M)^{\\frac{1}{3}} (R-t)^{\\frac{4}{3}}\n=\\frac{3}{4}(6M)^{\\frac{1}{3}} \\delta^{\\frac{4}{3}}\\leq 1.\n\\end{align*}\nFurthermore, \n$$|(Tg)(t_1)-(Tg)(t_2)|\\leq (6M\\delta)^{\\frac{1}{3}}|t_1-t_2|.$$\nThus, letting\n$$\\mathcal{A}=\\{g\\in \\mathcal{C}[R-\\delta, R]\\; :\\; g\\geq \\varphi,\\; \\max_{[R-\\delta, R]}|g|\\leq 1,\n\\; |g(t_1)-g(t_2)|\\leq (6M\\delta)^{\\frac{1}{3}}|t_1-t_2|\\quad\\forall\\, t_1, t_2\\in [R-\\delta, R]\\},$$\nwe note that $T:\\mathcal{A}\\to\\mathcal{A}$. Therefore, by Schauder fixed point theorem, \n$T$ has a fixed point $\\psi$ in $\\mathcal{A}$.\nIn particular, we get from \\eqref{ET2.2C}\n$$\\psi(t) = \\int_t^{R} \\left[\\int_{s}^{R} 6 e^{3K(s-\\zeta)} h(\\psi(\\zeta))\\, \\mathrm{d}\\zeta\\right]^{\\nicefrac{1}{3}} \\mathrm{d}{s}\\,.\n$$\nThis of course, implies $\\psi(R)=0$. Differentiating we obtain\n$$-(\\psi^\\prime(t))^3 = \\int_{t}^{R} 6 e^{3K(t-\\zeta)} h(\\psi(\\zeta))\\, \\mathrm{d}\\zeta\n\\quad t\\in (R-\\delta, R).$$\nThus $D_{-}\\psi(R)=0$ and differentiating the above equation we obtain\n\\begin{equation}\\label{ET2.2D}\n(\\psi^\\prime(t))^2\\psi^{\\prime\\prime}(t)-K(\\psi^\\prime(t))^3 - 2h(\\psi(t))=0\n\\quad \\text{for}\\; t\\in (R-\\delta, R).\n\\end{equation}\nExtend $\\psi$ in $(R, \\infty)$ be setting $\\psi(t)=0$ for $t\\geq R$. Note that\n$\\psi$ is continuously differentiable in $(R-\\delta, \\infty)$ and $\\psi'<0$ in\n$(R-\\delta, R)$.\n\nNow we let $\\kappa=\\frac{1}{8}$. Let $r_\\circ=R-\\delta-1$ and define\n $v(x)=\\psi(|x|+r_\\circ)$. Using \\eqref{ET2.2D} and the calculations in \\eqref{ET2.1E}\nwe see that for $ 1<|x|< 1+\\delta$ we have\n\\begin{align}\\label{ET2.2E}\n\\mathscr{L}_1 v + K |Dv|^3 - 2f(v)\n& =\n(\\psi^\\prime)^2(|x|+r_\\circ)\\psi^{\\prime\\prime}(|x|+r_\\circ)\n+ K |\\psi^{\\prime}|^3 - f(\\psi)\\nonumber\n\\\\\n&= (\\psi^\\prime)^2(|x|+r_\\circ)\\psi^{\\prime\\prime}(|x|+r_\\circ)\n- K (\\psi^{\\prime})^3 - 2h(\\psi)=0.\n\\end{align}\nWe claim that \n\\begin{equation}\\label{ET2.2F}\n\\mathscr{L}_1 v + K |Dv|^3 - f(v)=0 \\quad \\text{for}\\; |x|> 1\\,,\n\\end{equation}\nin viscosity sense. When $ 1<|x|< 1+ \\delta$, \\eqref{ET2.2F} follows\nfrom \\eqref{ET2.2E}. Again, for $|x|> 1+ \\delta$, \\eqref{ET2.2F} is evident.\nSo we consider the case where $|x|= 1+ \\delta$. Let $\\chi$ be a $\\mathcal{C}^2$ test with\n$v\\prec_{x}\\chi$. Since $v$ is $\\mathcal{C}^1$ we have $D\\chi(x)=Dv(x)=0$. Hence\n$$\\mathscr{L}_1\\chi(x) + K |D\\chi(x)|^3-f(v(x))=0,$$\nimplying $v$ is subsolution. Similarly, we show that $v$ is a supersolution. This \ngives us \\eqref{ET2.2F} and this completes the proof.\n\\end{proof}\n\n\n\n\\subsection*{Acknowledgement}\n\nThe author is grateful to the referee for his\/her careful reading and comments.\nThe research of Anup Biswas was supported in part by DST-SERB grants EMR\/2016\/004810 and MTR\/2018\/000028 and a SwarnaJayanti fellowship.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1}}%\n\n\\catcode`@=11\n\\long\\def\\@makecaption#1#2{\n   \\vskip 10pt\n   \\setbox\\@tempboxa\\hbox{{\\small\\bf #1.} \\ {\\small #2}}\n   \\ifdim \\wd\\@tempboxa >\\hsize      \n   {\\small\\bf #1.} \\ {\\small #2}\\par \n   \\else                             \n        \\hbox to\\hsize{\\hfil\\box\\@tempboxa\\hfil}\n   \\fi}\n\\catcode`@=12\n\n\n\\catcode`@=11\n\\def\\secteqno{\\@addtoreset{equation}{section}%\n\\def\\thesection.\\arabic{equation}}{\\thesection.\\arabic{equation}}}\n\\def\\endsecteqno{\\def\\thesection.\\arabic{equation}}{\\@ifundefined{chapter}%\n{\\arabic{equation}}{\\thechapter.\\arabic{equation}}}}\n\\newcounter{subequation}\n\\def\\alph{subequation}{\\alph{subequation}}\n\\def\\sneqnarray{\\stepcounter{equation}\\let\\@currentlabel=\\thesection.\\arabic{equation}}\n\\setcounter{subequation}{1}\n\\def\\@eqnnum{{\\rm (\\thesection.\\arabic{equation}}\\alph{subequation})}}\n\\global\\@eqcnt\\z@\\tabskip\\@centering\\let\\\\=\\@eqncr\\let\\@@eqncr=\\@@sneqncr\n$$\\halign to \\displaywidth\\bgroup\\@eqnsel\\hskip\\@centering\n $\\displaystyle\\tabskip\\z@{##}$&\\global\\@eqcnt\\@ne\n \\hskip 2\\arraycolsep \\hfil${##}$\\hfil\n &\\global\\@eqcnt\\tw@ \\hskip 2\\arraycolsep\n$\\displaystyle\\tabskip\\z@{##}$\\hfil\ntabskip\\@centering&\\llap{##}\\tabskip\\z@\\cr}\n\\def\\@@sneqncr\\egroup $$\\global\\@ignoretrue{\\@@sneqncr\\egroup $$\\global\\@ignoretrue}\n\\def\\@@sneqncr{\\let\\@tempa\\relax\n   \\ifcase\\@eqcnt \\def\\@tempa{& & &}\\or \\def\\@tempa{& &}\n   \\else \\def\\@tempa{&}\\fi\n     \\@tempa \\if@eqnsw\\@eqnnum\\stepcounter{subequation}\\fi\n     \\global\\@eqnswtrue\\global\\@eqcnt\\z@\\cr}\n\\def\\def\\@lbibitem[##1]##2{\\@bibitem{##2}}{\\def\\@lbibitem[##1]##2{\\@bibitem{##2}}}\n\\catcode`@=12\n\n\n\\def\\begin{equation}{\\begin{equation}}\n\\def\\end{equation}{\\end{equation}}\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\\def\\nonumber{\\nonumber}\n\n\n\\def\\alpha}  \\def\\g{\\gamma} \\def\\G{\\Gamma} \\def\\lapp{\\lambda^{\\prime\\prime}{\\alpha}  \\def\\g{\\gamma} \\def\\G{\\Gamma} \\def\\lapp{\\lambda^{\\prime\\prime}}\n\\def\\lambda} \\def\\lap{\\lambda^{\\prime}} \\def\\pa{\\partial} \\def\\de{\\delta} \\def\\De{\\Delta} \\def\\dag{\\dagger{\\lambda} \\def\\lap{\\lambda^{\\prime}} \\def\\pa{\\partial} \\def\\de{\\delta} \\def\\De{\\Delta} \\def\\dag{\\dagger}\n\\def\\epsilon} \\def\\nb{\\bm{\\nabla}}  \\def\\Oc{{\\rm O}} \\def\\S{{\\rm S}{\\epsilon} \\def\\nb{\\bm{\\nabla}}  \\def\\Oc{{\\rm O}} \\def\\S{{\\rm S}}\n\\def{\\bm \\nabla}{{\\bm \\nabla}}\n\\def{\\bm \\sigma}{{\\bm \\sigma}}\n\\def\\overleftrightarrow{\\partial}{\\overleftrightarrow{\\partial}}\n\\def\\Lambda_{\\rm QCD}{\\Lambda_{\\rm QCD}}\n\\newcommand{\\tensor}[1]{\\ensuremath{\\underline{\\mathbf{#1}}}}\n\\newcommand{\\Tensor}[1]{\\ensuremath{\\underline{\\boldsymbol{#1}}}}\n\n\n\\documentclass[aps,10pt,prd,showpacs,amsmath,amssymb,preprintnumbers,superscriptaddress,nofootinbib,showkeys]{revtex4-1}\n\n\\usepackage[german,english]{babel}\n\\usepackage{hyperref}\n\\usepackage{amssymb}\n\\usepackage{amsmath}\n\\usepackage{bm\n\\usepackage{braket}\n\\usepackage{slashed}\n\\usepackage{multirow}\n\\usepackage{epsfig}\n\\usepackage{epstopdf}\n\\usepackage{bbm}\n\\usepackage{bbold}\n\\usepackage{soul\n\n\\DeclareMathOperator\\arctanh{arctanh}\n\\pdfoptionpdfminorversion=7\n\\pdfsuppresswarningpagegroup=1\n\n\n\\begin{document}\n\n\\title{Effective QCD string and doubly heavy baryons}\n\\author{Joan Soto}\n\\email{joan.soto@ub.edu}\n\\affiliation{Departament de F\\'\\i sica Qu\\`antica i Astrof\\'isica and Institut de Ci\\`encies del Cosmos, Universitat de Barcelona (IEEC-UB), Mart\\'\\i$\\,$ i Franqu\\`es 1, 08028 Barcelona, Catalonia, Spain}\n\n\\author{Jaume Tarr\\'us Castell\\`a}\n\\email{jtarrus@ifae.es}\n\\affiliation{Grup de F\\'\\i sica Te\\`orica, Departament de F\\'\\i sica and IFAE-BIST, Universitat Aut\\`onoma de Barcelona,\\\\ \nE-08193 Bellaterra (Barcelona), Catalonia, Spain}\n\n\\date{\\today}\n\n\\begin{abstract}\nExpressions for the potentials appearing in the nonrelativistic effective field theory description of doubly heavy baryons are known in terms of operator insertions in the Wilson loop. However, their evaluation requires nonperturbative techniques, such as lattice QCD, and the relevant calculations are often not available. We propose a parametrization of these potentials with a minimal model dependence based on an interpolation of the short- and long-distance descriptions. The short-distance description is obtained from weakly-coupled potential NRQCD and the long-distance one is computed using an effective string theory. The effective string theory coincides with the one for pure gluodynamics with the addition of a fermion field constrained to move on the string. We compute the hyperfine contributions to the doubly heavy baryon spectrum. The unknown parameters are obtained from heavy quark-diquark symmetry or fitted to the available lattice-QCD determinations of the hyperfine splittings. Using these parameters we compute the double charm and bottom baryon spectrum including the hyperfine contributions. We compare our results with those of other approaches and find that our results are closer to lattice-QCD determinations, in particular for the excited states. Furthermore, we compute the vacuum energy in the effective string theory and show that the fermion field contribution produces the running of the string tension and a change of sign in the L\\\"uscher term.\n\\end{abstract}\n\n\\maketitle\n\n\n\\section{Introduction}\n\nThe discovery of more than two dozen exotic quarkonium states, as well as the more recent measurements of pentaquarks and double charm baryons, has increased interest in the wider class of hadrons containing two heavy quarks. All doubly heavy hadrons have in common that the constituent heavy quarks are nonrelativistic and that the dynamics of the heavy quarks and the light degrees of freedom, light quarks and gluons, can be factorized in an adiabatic expansion. An effective field theory (EFT) for doubly heavy hadrons built upon these two expansions was presented in Ref.~\\cite{Soto:2020xpm}. Since the EFT reproduces the Born-Oppenheimer (BO) approximation at leading order we will refer to it as BOEFT. In the construction of the EFT no assumption is made about the heavy-quark distance and hence the EFT is valid both for short and long distances with respect to $\\Lambda^{-1}_{\\rm QCD}$, the inverse of the intrinsic scale of the nonperturbative effects in QCD. Therefore, the EFT can be seen as a generalization of strongly coupled potential NRQCD (pNRQCD)~\\cite{Brambilla:2000gk,Pineda:2000sz} for quarkonium states to any heavy-quark-pair state with nontrivial light degrees of freedom. The matching coefficients of BOEFT depend on the heavy-quark-pair distance and, therefore, correspond to potential interactions. Expressions for these potentials in terms of operator insertions in the Wilson loop can be obtained by matching BOEFT to NRQCD~\\cite{Caswell:1985ui,Bodwin:1994jh,Manohar:1997qy}, which can also be found in Ref.~\\cite{Soto:2020xpm}. Since the Wilson loops involve nonperturbative dynamics, in principle they should be evaluated with lattice QCD.\n\nBOEFT has been applied to doubly heavy baryons in Ref.~\\cite{Soto:2020pfa}. In this case, the Wilson loop with light quark operator insertions corresponding to the static potential, has been obtained in the lattice~\\cite{Najjar:2009da,Najjarthesis} including several excited states. This lattice data was used in Ref.~\\cite{Soto:2020pfa} to obtain the double charm and bottom baryon spectrum at leading order in BOEFT. The leading-order spectrum is formed by spin-symmetry multiplets of states with total angular momentum $j$ and parity ${\\eta_p}$. The degeneracy of the states in the multiplets is broken by $1\/m_Q$ suppressed operators in BOEFT, where $m_Q$ is the heavy-quark mass. These operators correspond to different couplings of the heavy-quark spin and angular momentum to the light-quark spin. Unfortunately, at the moment there is no lattice data available for the potentials of these heavy-quark spin and angular-momentum dependent operators.\n\nThe main aim of this paper is to develop a parametrization for the subleading potentials for doubly heavy baryons. The short-distance regime is defined as $r\\ll 1\/\\Lambda_{\\rm QCD}$, which is equivalent to assuming that there is an energy gap between the relative momentum of the heavy quarks $m_Q v$, with $v$ the relative velocity, and $\\Lambda_{\\rm QCD}$. Therefore, in the short-distance regime one can build BOEFT in two steps. First, the relative momentum is integrated out perturbatively in order to build weakly-coupled pNRQCD~\\cite{Pineda:1997bj,Brambilla:1999xf,Brambilla:2005yk}, and second, one integrates out the $\\Lambda_{\\rm QCD}$ modes. This procedure results in multipole expanded expressions of the potentials in BOEFT where the dependence on the heavy-quark-pair distance is explicit and the nonperturbative dynamics is encoded in some unknown constants. Examples of this two-step matching can be found in Refs.~\\cite{Brambilla:2018pyn,Brambilla:2019jfi} for the heavy-quark spin dependent potentials of quarkonium hybrids and in Refs.~\\cite{Pineda:2019mhw,Castella:2021hul} for the hybrid to standard quarkonium transitions. \n\nIn the long-distance regime, $r\\gg 1\/\\Lambda_{\\rm QCD}$, it is known that the heavy-quark-antiquark static potential obtained from lattice QCD is well described in terms of an Effective String Theory (EST)~\\cite{Nambu:1978bd} modeling the flux tube formed between the heavy-quark-antiquark pair at large separations. Corrections to the long-distance linear behavior of the static potential can be calculated in a systematic manner in the EST~\\cite{Luscher:1980fr,Luscher:2002qv} (see also \\cite{Polchinski:1991ax,Aharony:2013ipa}), including the contribution from the vacuum energy of the string which has also been confirmed by lattice QCD~\\cite{Luscher:2002qv,Juge:2002br,Brandt:2017yzw}. The long-distance behavior of the subleading potentials for quarkonium can be computed in the EST given a mapping of the Wilson loop with operator insertions into EST correlation functions. This mapping was worked out in Ref.~\\cite{PerezNadal:2008vm} and some of the subleading potentials were computed. This computation was later extended up to next-to-leading order in the EST in Refs.~\\cite{Brambilla:2014eaa,Hwang:2018rju}. The parametrization given by these computations agree well with the lattice determinations of Refs.~\\cite{Koma:2006si,Koma:2006fw}. The excitations of the string produce a spectrum of excited states, corresponding to quarkonium hybrid static potentials, which accurately describe the lattice determinations at long distances~\\cite{Juge:2002br}. The mapping of operators to the EST to compute subleading potentials for hybrid quarkonium was introduced in Ref.~\\cite{Oncala:2017hop}.\n\nIn this paper we present an EST for two static heavy quarks and one valence light quark, which is suitable to compute the long-distance part of the potentials of BOEFT for doubly heavy baryons. We obtain the mapping between different operator insertions in the Wilson loop and correlators in the EST and use it to compute the static potential and the heavy-quark spin and angular-momentum dependent potentials in the long-distance regime. A parametrization of the potentials for any distance between the heavy-quark pair is built by interpolating between the short- and long-distance descriptions. The free parameters of the short- and long-distance descriptions of the potentials are then fitted to a broad set of lattice data on the hyperfine splittings of doubly heavy baryons~\\cite{Briceno:2012wt,Namekawa:2013vu,Brown:2014ena,Alexandrou:2014sha,Bali:2015lka,Padmanath:2015jea,Alexandrou:2017xwd,Lewis:2008fu,Brown:2014ena,Mohanta:2019mxo,Bahtiyar:2020uuj}. Using this parametrization of the potentials we compute the hyperfine contributions to the double charm and bottom baryons states of Ref.~\\cite{Soto:2020pfa} corresponding to spin $1\/2$ light-quark states. These include all the states below threshold of double bottom baryons, for which no lattice determination exists beyond the ground state spin doublet. Finally, we compare our results with previous model based determinations of the masses of doubly heavy baryons.\n\nWe present the paper as follows. In Sec.~\\ref{sec:dhbp}, we review the general structure of the doubly heavy baryon potentials at next-to-leading order in the $1\/m_Q$ expansion. We also discuss the short-distance constraints for those potentials. The leading-order parameters can be extracted from the heavy-light meson spectrum using heavy quark-diquark symmetry. In Sec.~\\ref{sec:est}, we propose an EST with fermionic degrees of freedom in order to describe the long-distance behavior of the potentials. Based on the $D_{\\infty h}$ group, we put forward a mapping from the NRQCD operator insertions in the Wilson loop to EST operators, and use it to compute the potentials. At leading order, they turn out to depend on two parameters only. In Sec.~\\ref{hysc} we review the expressions of the doubly heavy baryon hyperfine splittings. In Sec.~\\ref{sec:intfp} we model the spin-dependent potentials using suitable interpolations between the known short-distance behavior and the just calculated long-distance one. Then, the remaining unknown parameters of the parametrization of the potentials are fitted to lattice data on the hyperfine splittings of doubly heavy baryons. Using these parameters, we predict the spectrum of doubly heavy baryons including hyperfine contributions in Sec.~\\ref{sec:dhbs}. We compare our results with other approaches in Sec.~\\ref{sec:models}. We close the paper with some conclusions in Sec~\\ref{sec:con}. In Appendix~\\ref{app:ce} we calculate the Casimir energy (L\\\"uscher term) in the EST. Finally, in Appendix~\\ref{app:sdec} we give expressions for the short-distance regime constants as correlators in weakly-coupled pNRQCD.\n\n\\section{Doubly heavy baryon potentials}\\label{sec:dhbp}\n\n\\subsection{General expressions}\n\nA general EFT framework to describe any doubly heavy hadron has been presented in Ref.~\\cite{Soto:2020xpm}. The EFT was worked out up to $1\/m_Q$ including the terms that depend on the heavy-quark spin and angular momentum. The matching expressions of the potentials in terms of operator insertions in the Wilson loop can also be found in Ref.~\\cite{Soto:2020xpm}. This EFT framework has been applied to doubly heavy baryons in Ref.~\\cite{Soto:2020pfa} where the spectrum associated to the four lowest lying static energies was obtained. These static energies are characterized by the representation of $D_{\\infty h}$ and the quantum numbers of the light-quark operator that interpolates them, in particular the spin $\\kappa$ and parity $p$. In the present work we will only consider the cases with light-quark interpolating operator with $\\kappa^p=(1\/2)^{\\pm}$. These spin-$1\/2$ cases only have one possible projection into the heavy-quark axis and therefore each correspond to a single $D_{\\infty h}$ representation. These are $(1\/2)_g$ and $(1\/2)_u'$ for the $\\kappa^p=(1\/2)^{+}$ and $\\kappa^p=(1\/2)^{-}$ operators, respectively.\n\nThe Hamiltonian densities associated to the $\\kappa^p=(1\/2)^{\\pm}$ light-quark states~\\cite{Soto:2020pfa} have the following expansion up to $1\/m_Q$\n\\begin{align}\nh_{(1\/2)^{\\pm}}=\\frac{\\bm{p}^2}{m_Q}+\\frac{\\bm{P}^2}{4m_Q}+V_{(1\/2)^{\\pm}}^{(0)}(\\bm{r})+\\frac{1}{m_Q}V_{(1\/2)^{\\pm}}^{(1)}(\\bm{r},\\,\\bm{p})\\,.\\label{hamden}\n\\end{align}\n At leading order we have just the static potential\n\\begin{align}\nV_{(1\/2)^{\\pm}}^{(0)}(\\bm{r})=&V_{(1\/2)^{\\pm}}^{(0)}(r)\\,.\\label{lopot}\n\\end{align}\nThe heavy-quark spin and angular-momentum dependent operators appear at next-to-leading order and read as\n\\begin{align}\nV_{(1\/2)^{\\pm}{\\rm SD}}^{(1)}(\\bm{r})=&V^{s1}_{(1\/2)^{\\pm}}(r)\\bm{S}_{QQ}\\cdot\\bm{S}_{1\/2}+V^{s2}_{(1\/2)^{\\pm}}(r)\\bm{S}_{QQ}\\cdot\\left(\\bm{{\\cal T}}_{2}\\cdot\\bm{S}_{1\/2}\\right)+V^{l}_{(1\/2)^{\\pm}}(r)\\left(\\bm{L}_{QQ}\\cdot\\bm{S}_{1\/2}\\right)\\,,\\label{nlopot}\n\\end{align}\nwith ${\\cal T}^{ij}_2=\\hat{\\bm{r}}^i\\hat{\\bm{r}}^j-\\delta^{ij}\/3$, $\\bm{S}_{1\/2}=\\bm{\\sigma}\/2$ and $2\\bm{S}_{QQ}=\\bm{\\sigma}_{QQ}=\\bm{\\sigma}_{Q_1}\\mathbb{1}_{2\\,Q_2}+\\mathbb{1}_{2\\,Q_1}\\bm{\\sigma}_{Q_2}$, where $\\bm{\\sigma}$ are the standard Pauli matrices and $\\mathbb{1}_2$ is an identity matrix in the heavy-quark spin space for the heavy quark labeled in the subindex.\n\nThe matching expressions of the potentials in terms of operator insertions in the Wilson loop can be found in Ref.~\\cite{Soto:2020xpm}. For the potentials in Eqs.~\\eqref{lopot} and \\eqref{nlopot} the expressions in Ref.~\\cite{Soto:2020xpm} reduce to\n\\begin{align}\nV_{(1\/2)^{\\pm}}^{(0)}(\\bm{r})&=\\lim_{t\\to\\infty}\\frac{i}{t}\\log\\left({\\rm Tr}\\left[\\langle1\\rangle^{(1\/2)^{\\pm}}_{\\Box}\\right]\\right)\\,,\\label{lopotst}\n\\end{align}\nand\n\\begin{align}\nV^{s1}_{(1\/2)^{\\pm}}(r)&=-c_F\\lim_{t\\to\\infty}\\frac{4}{3t}\\int^{t\/2}_{-t\/2} dt^{\\prime}\\frac{{\\rm Tr}\\left[\\bm{S}_{1\/2}\\cdot\\langle g\\bm{B}(t^{\\prime},\\bm{x}_1)\\rangle^{(1\/2)^{\\pm}}_{\\Box}\\right]}{{\\rm Tr}\\left[\\langle1\\rangle^{(1\/2)^{\\pm}}_{\\Box} \\right]}\\,,\\label{nlopots1}\\\\\nV^{s2}_{(1\/2)^{\\pm}}(r)&=-c_F\\lim_{t\\to\\infty}\\frac{6}{t}\\int^{t\/2}_{-t\/2} dt^{\\prime}\\frac{{\\rm Tr}\\left[\\left(\\bm{S}_{1\/2}\\cdot \\bm{{\\cal T}}_{2}\\right)\\cdot\\langle g\\bm{B}(t^{\\prime},\\bm{x}_1)\\rangle^{(1\/2)^{\\pm}}_{\\Box}\\right]}{{\\rm Tr}\\left[\\langle1\\rangle^{(1\/2)^{\\pm}}_{\\Box} \\right]}\\,,\\label{nlopots2}\\\\\nV^{l}_{(1\/2)^{\\pm}}=&-\\lim_{t\\to\\infty}2\\int^{1}_{0} ds\\,s\\frac{{\\rm Tr}\\left[\\bm{S}_{1\/2}\\cdot\\left(\\frac{2}{3}\\mathbb{1}_2 -\\bm{{\\cal T}}_{2}\\right)\\cdot\\langle g\\bm{B}(t\/2,\\bm{z}(s))\\rangle^{(1\/2)^{\\pm}}_{\\Box}\\right]}{{\\rm Tr}\\left[\\langle1\\rangle^{(1\/2)^{\\pm}}_{\\Box} \\right]}\\,,\\label{nlopotl}\n\\end{align}\nwhere $\\bm{z}(s)=\\bm{x}_1+s(\\bm{R}-\\bm{x}_1)$ and we use the following notation for the Wilson loop averages\n\\begin{align}\n&\\langle \\dots\\rangle^{(1\/2)^{\\pm}}_{\\Box}=\\langle {\\cal Q}_{(1\/2)^{\\pm}}(t\/2,\\,\\bm{R})\\dots {\\cal Q}_{(1\/2)^{\\pm}}^{\\dagger}(-t\/2,\\,\\bm{R})P\\left\\{e^{-ig\\int_{{\\cal C}_1+{\\cal C}_2}dz^{\\mu}A^{\\mu}(z)}\\right\\}\\rangle\\,,\n\\end{align}\nwith ${\\cal C}_1$ and ${\\cal C}_2$ the upper and lower paths of a rectangular Wilson loop. Note that, unlike the quark-antiquark case, the flow is in the same direction for both paths. The interpolating operators are\n\\begin{align}\n{\\cal Q}^{\\alpha}_{(1\/2)^+}(t,\\bm{x})&=\\left[P_+q^{l}(t,\\bm{x})\\right]^\\alpha\\underline{T}^l \\,,\\\\ \n{\\cal Q}^{\\alpha}_{(1\/2)^-}(t,\\bm{x})&=\\left[P_+\\gamma^5q^l(t,\\bm{x})\\right]^\\alpha\\underline{T}^l \\,,\n\\end{align}\nwhere $\\alpha=-1\/2,1\/2$, and we have used the following $\\bar{3}$ tensor invariants \n\\begin{align}\n&\\underline{T}^l_{ij}  = \\frac{1}{\\sqrt{2}} \\epsilon_{lij},\\quad i,\\,j,\\,l=1,2,3\\,.\n\\end{align}\n\n\\subsection{Short-distance potentials}\\label{sec:sdp}\n\nThe short-distance regime is characterized by $r\\ll \\Lambda^{-1}_{\\rm QCD}$. Since, in this regime the heavy-quark-pair distance and $\\Lambda^{-1}_{\\rm QCD}$ are well-separated scales the matching of NRQCD to the BOEFT for doubly heavy baryons can be done in two steps. First, one integrates out the heavy-quark-pair distance, which can be done in perturbation theory. This produces weakly-coupled potential NRQCD (pNRQCD) for doubly heavy systems presented in Ref.~\\cite{Brambilla:2005yk}. Then, integrating out the $\\Lambda_{\\rm QCD}$ modes one recovers BOEFT. This procedure delivers expressions of the potentials in Eqs.~\\eqref{lopotst}-\\eqref{nlopotl} as an expansion in the heavy-quark-pair distance. An analogous approach was used in Refs.~\\cite{Brambilla:2018pyn,Brambilla:2019jfi} to determine short distance expansion of the hybrid quarkonium potentials. All the potentials follow the same general structure in the short-distance regime; a possible nonanalytic term in $r$ produced by integrating out the heavy-quark-pair distance and an expansion in powers of $r^2$ with nonperturbative coefficients. These nonperturbatice coefficients only depend on the $\\Lambda_{\\rm QCD}$ scale and can be expressed as weakly-coupled pNRQCD correlators of light quark and gluon operators.\n\n\\begin{figure}[ht!]\n   \\centerline{\\includegraphics[width=.9\\textwidth]{sdm_stp.pdf}}\n\t\\caption{Matching of the Wilson loop for the static potential for doubly heavy baryons, with the expansion in weakly-coupled pNRQCD up to next-to-leading order. The single lines represent the antitriplet fields, the double lines the sextet field, the dotted and the curly lines the light-quark and transverse gluon fields respectively (emissions of longitudinal gluon fields from the triplet and sextet fields and from the vertices are omitted). The crossed circles indicate the insertion of a ${\\cal Q}$ operator and the square or diamond the insertion of a chromoelectric dipole or quadrupole operator, respectively.}\n\t\\label{sp_matching}\n \\end{figure}\n\nThe expansion of the static potential in Eq.~\\eqref{lopotst} is given diagrammatically in Fig.~\\ref{sp_matching} and corresponds to the following form\n\\begin{align}\nV^{(0)}_{(1\/2)^{\\pm}}(r)&=-\\frac{2}{3}\\frac{\\alpha_s}{r}+\\overline{\\Lambda}_{(1\/2)^{\\pm}}+\\overline{\\Lambda}^{(1)}_{(1\/2)^{\\pm}}r^2+\\dots\\,,\\label{sdstp}\n\\end{align}\nwith the nonperturbative constants given as pNRQCD correlators in Appendix~\\ref{app:sdec}.\n\n\nFor the heavy-quark spin and angular-momentum dependent potentials the short-distance expansion of the potentials is given diagrammatically in Fig.~\\ref{sdp_matching} and are as follows:\n\n\\begin{figure}[ht!]\n   \\centerline{\\includegraphics[width=.99\\textwidth]{sdm_sdp.pdf}}\n\t\\caption{Matching of the heavy-quark spin dependent potentials up to next-to-leading order in weakly-coupled pNRQCD. The legend is as in Fig.~\\ref{sp_matching} with the addition of the solid dot, a white-dotted square, and a white-dotted diamond representing the insertion of a leading-order, dipole, and quadrupole heavy-quark spin chromomagnetic couplings, respectively. Further next to leading diagrams can be generated by changing the order of the different internal vertices, and by adding an extra transverse gluon emission to the heavy-quark spin chromomagnetic couplings. The potential of the heavy-quark angular-momentum dependent operator is matched to an analogous expansion.}\n\t\\label{sdp_matching}\n \\end{figure}\n\n\\begin{align}\nV^{s1}_{(1\/2)^{\\pm}}(r)&=c_F\\left(\\Delta^{(0)}_{(1\/2)^{\\pm}}+\\Delta^{(1,0)}_{(1\/2)^{\\pm}}r^2+\\dots\\right)\\,,\\label{sdps1}\\\\\nV^{s2}_{(1\/2)^{\\pm}}(r)&=c_F\\Delta^{(1,2)}_{(1\/2)^{\\pm}}r^2+\\dots\\,,\\label{sdps2}\\\\\nV^{l}_{(1\/2)^{\\pm}}=&\\frac{1}{2}\\left[\\Delta^{(0)}_{(1\/2)^{\\pm}}+\\left(\\Delta^{(1,0)}_{(1\/2)^{\\pm}}-\\frac{1}{3}\\Delta^{(1,2)}_{(1\/2)^{\\pm}}\\right)r^2\\right]+\\dots\\,,\\label{sdpl}\n\\end{align}\nwith the nonperturbative constants given in Appendix~\\ref{app:sdec}. At leading order both Eqs.~\\eqref{sdps1} and \\eqref{sdpl} depend on the same correlator and the difference in the contribution to the potential stems from different factors in the coupling of the heavy-quark spin and angular momentum to the chromomagnetic field in the Lagrangian of Eq.~(9) in Ref.~\\cite{Brambilla:2005yk}. The potential of the spin-tensor coupling in the Lagrangian of Eq.~\\eqref{nlopot} vanishes at leading order since, to appear, it requires the insertion in the pNRQCD correlator of operators carrying the dependence on $\\bm{r}$ which are suppressed in the multipole expansion. This type of correlator is also responsible for the next-to-leading order contributions to Eqs.~\\eqref{sdps1} and \\eqref{sdpl}. It is interesting to note, that the next-to-leading coefficient of Eq.~\\eqref{sdpl} can be written as a combination of the next-to-leading coefficients of Eqs.~\\eqref{sdps1} and \\eqref{sdps2}.\n\nIn the static and $r\\to 0$ limits the heavy-quark pair becomes indistinguishable from a single heavy antiquark. This is the so-called heavy quark-diquark duality~\\cite{Savage:1990di,Hu:2005gf,Mehen:2017nrh,Mehen:2019cxn}. One can use this duality to relate the leading-order coefficients of the expansions of the potentials in Eqs.~\\eqref{sdstp}-\\eqref{sdpl} to the heavy meson masses. The value of $\\overline{\\Lambda}_{(1\/2)^{+}}$ is equal to the leading-nonperturbative contribution to the lowest laying D- or B-meson masses, usually referred to as just $\\overline{\\Lambda}$.\n\nThe value of $\\overline{\\Lambda}$ has been obtained in Refs.~\\cite{Bazavov:2018omf,Ayala:2019hkn} combining lattice determinations of the heavy-meson masses and perturbative computations of the heavy-quark masses. It is therefore necessary to use values of $\\overline{\\Lambda}$ and the heavy-quark masses computed in the same scheme. We use the values of Ref.~\\cite{Bazavov:2018omf} in the MRS scheme\n\\begin{align}\n&m_c=1.392(11)~{\\rm GeV}\\,, \\label{mc}\\\\\n&m_b=4.749(18)~{\\rm GeV}\\,, \\label{mb}\\\\\n&\\overline{\\Lambda}_{(1\/2)^+}=0.555(31)~{\\rm GeV} \\,.\\label{lbar12}\n\\end{align}\nFollowing from the heavy quark-diquark duality, the difference $\\overline{\\Lambda}_{(1\/2)^{-}}-\\overline{\\Lambda}_{(1\/2)^{+}}$ is equal to the mass gap between the ground and first excited heavy-light mesons, up to corrections of order $\\Lambda^2_{\\rm QCD}\/m_Q$. The values for this difference are collected in Table~\\ref{ldt}. The values are compatible with the short-distance energy gaps between the static energies $(1\/2)_g$ and $(1\/2)_u'$ of Refs.~\\cite{Najjar:2009da,Najjarthesis} associated to the light-quark operators $(1\/2)^{+}$ and $(1\/2)^{-}$, respectively.\n\n\n\\begin{table}[ht!]\n\\begin{tabular}{||cc||}\\hline\\hline\n{\\rm Heavy Mesons} & $(\\overline{\\Lambda}_{(1\/2)^{-}}-\\overline{\\Lambda}_{(1\/2)^{+}})[{\\rm GeV}]$ \\\\ \\hline\n$m_{D^*_0(2300)^0}-m_{D^0}$              & $0.435(19)$ \\\\\n$m_{D^*_0(2300)^\\pm}-m_{D^\\pm}$          & $0.465(7)$ \\\\\n$m_{D_1(2420)^0}-m_{D^{*}(2007)^0}$      & $0.41375(7)$ \\\\\n$m_{D_1(2420)^\\pm}-m_{D^{*}(2010)^\\pm}$  & $0.4129(24)$ \\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{Determination of $\\overline{\\Lambda}_{(1\/2)^{-}}-\\overline{\\Lambda}_{(1\/2)^{+}}$ from D meson mass differences. The masses are taken from the PDG~\\cite{Zyla:2020zbs}. The uncertainty corresponds only to the experimental uncertainty of the meson masses. The uncertainty in the determination of $(\\overline{\\Lambda}_{(1\/2)^{-}}-\\overline{\\Lambda}_{(1\/2)^{+}})$ due to neglected higher-order terms is expected to be about $30\\%$.}\n\\label{ldt} \n\\end{table}\n\nFinally, the value of $\\Delta^{(0)}_{(1\/2)^{\\pm}}$ can be related to the hyperfine splittings in D or B mesons~\\cite{Brambilla:2005yk}\n\\begin{align}\n&m_{P^*_{\\bar{Q}q}}-m_{P_{\\bar{Q}q}}=\\frac{2c_F(m_Q)}{m_Q}\\Delta^{(0)}_{(1\/2)^{\\pm}}\\,,\\label{hmhf}\n\\end{align}\nwith corrections expected to be of order $\\Lambda^3_{\\rm QCD}\/m^2_Q$. The values of $\\Delta^{(0)}_{(1\/2)^{\\pm}}$ from Eq.~\\eqref{hmhf} for various heavy-meson masses are found in Table~\\ref{delahmm}.\n\n\\begin{table}[ht!]\n\\begin{tabular}{||cc||}\\hline\\hline\n{\\rm Heavy Mesons} & $\\Delta^{(0)}_{(1\/2)^{+}}~[{\\rm GeV}^2]$ \\\\ \\hline\n\n$m_{D^*(2007)^{0}}-m_{D^0}$ & $0.08819(2)$ \\\\\n\n$m_{D^*(2010)^{\\pm}}-m_{D^{\\pm}}$ & $0.087317(9)$ \\\\\n\n$m_{B^{0*}}-m_{B^0}$ & $0.1222(6)$ \\\\\n\n$m_{B^{\\pm*}}-m_{B^{\\pm}}$ & $0.1226(6)$ \\\\ \\hline\\hline\n{\\rm Heavy Mesons} &  $\\Delta^{(0)}_{(1\/2)^{-}}~[{\\rm GeV}^2]$  \\\\ \\hline\n$m_{D_1(2420)^0}-m_{D^*_0(2300)^0}$ & 0.075(11) \\\\\n\n$m_{D_1(2420)^\\pm}-m_{D^*_0(2300)^\\pm}$ & 0.0461(3)\\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{Determination of $\\Delta^{(0)}_{(1\/2)^{\\pm}}$ from the heavy meson masses, taken from the PDG~\\cite{Zyla:2020zbs}, using Eq.~\\eqref{hmhf}. We take the renormalization group improved expression for $c_F$ at $1$~GeV. The uncertainty corresponds only to the experimental uncertainty of the meson masses. The uncertainty in the determination of $\\Delta^{(0)}_{(1\/2)^{\\pm}}$ due to neglected higher order terms is expected to be of $\\sim 30\\%$ for the charm mesons and $\\sim 10\\%$ for the bottom mesons.}\n\\label{delahmm}\n\\end{table}\n\n\\section{Effective string theory}\\label{sec:est}                                                                                                                                                \n\\subsection{Motivation}\\label{est:m}\n\nThe QCD potentials for heavy quarks can be calculated assuming the heavy quarks to be static color sources. For a heavy-quark-antiquark system, the leading-order (static) potential is the energy of a source in the fundamental representation and a source in the complex conjugate representation separated at a distance $r$. Since the system must be a color singlet object, a certain gluon configuration must exist between the two sources in order to achieve so. When the distance is larger than the typical QCD scale $r\\Lambda_{\\rm QCD} \\gg 1$, a flux tube emerges \\cite{Bali:1994de}, with a typical radius $\\sim \\Lambda_{\\rm QCD}^{-1}$. Assuming a constant energy per unit length in the flux tube leads to a linear potential. The flux-tube dynamics can be described by an EST, which matches the lattice QCD calculations very well for the static potential at long distances in the absence of light quarks \\cite{Luscher:2002qv,Juge:2002br,Brandt:2017yzw}. When light quarks are present, the flux-tube configuration is still observed \\cite{Ichie:2002dy} even though it may break due to light quark-antiquark pair creation, a phenomenon known as string breaking \\cite{Bali:2005fu,Bulava:2019iut}. Nevertheless a flux-tube like configuration leading to a linear potential remains as an excited state for $r$ beyond the string-breaking scale.\n\nFor a baryon with two heavy quarks, we have an analogous situation. The two sources are now in the fundamental representation, and the gluon configuration linking them must also contain a valence light quark. When $r\\Lambda_{\\rm QCD} \\gg 1$ we expect a flux tube to emerge from each source and to joint at the point between them where the valence light quark is at each time. Hence, the naive expectation would be to have a potential with the same string tension as in the quark-antiquark system plus a constant contribution $\\sim \\Lambda_{\\rm QCD}$ due to the extra energy provided by the link to the valence light quark. Lattice QCD simulations indeed observe a linear potential \\cite{Yamamoto:2008jz,Najjar:2009da}. Hence, we expect an EST to account for the long-distance behavior of the potential as well. Locally, the EST should be the same as the one for the quark-antiquark system, but it must contain some additional degrees of freedom describing the link to the valence light quark. In particular it must keep its transformation properties under $D_{\\infty h}$ and flavor. We propose to add a fermion to the usual EST which transforms like the light quark under flavor and the Lorentz group. We write down a reparametrization invariant Lagrangian, and expand it at the desired order in the effective theory expansion. \n\n\n\\subsection{Construction}\\label{est:c}\n\nA string has one spatial dimension and its motion through spacetime defines a world sheet. The world sheet can be parametrized with two variables, which we will denote by $x=(\\tau,\\,\\lambda)$. The embedding of the world-sheet in Minkowsky space is given by\n\\begin{align}\n\\bm{\\xi}=\\left(\\xi^0(\\tau,\\lambda),\\,\\xi^1(\\tau,\\lambda),\\,\\xi^2(\\tau,\\lambda),\\,\\xi^3(\\tau,\\lambda)\\right)\\,.\n\\end{align}\nThe metric $g_{ab}$ induced on the string reads\n\\begin{align}\ng_{ab}=\\eta_{\\alpha\\beta}e^{\\alpha}_ae^{\\beta}_b\\,,\n\\end{align}\nwith $\\eta_{\\alpha\\beta}$ the Minkowsky metric, and $e^{\\alpha}_a\\equiv\\partial \\xi^{\\alpha}\/\\partial x^a$ the Zweibein. The action of the gluonic string is just proportional to the area of the string world sheet\n\\begin{align}\nS_{\\rm g}=-\\sigma\\int d^2 x\\sqrt{g}\\,,\\label{gst}\n\\end{align}\nwith $\\sigma$ the string tension and $g=|$det$g_{ab}|$.\n\nThe action of a four-dimensional Dirac field constrained on a string is given by\n\\begin{align}\nS_{\\rm l.q}=\\int d^2 x\\sqrt{g}\\bar{\\psi}(x)\\left(i\\rho^a\\overleftrightarrow{\\partial}_a-m_{\\rm l.q.}\\right)\\psi(x)\\quad,\\quad \\bar{\\psi}\\rho^a\\overleftrightarrow{\\partial}_a\\psi\\equiv\\left(\\bar{\\psi}(\\rho^a\\pa_a\\psi)-(\\pa_a\\bar{\\psi})\\rho^a\\psi\\right)\/2\n\\label{ngaplq}\n\\end{align}\nwith $\\rho^a\\equiv \\gamma^{\\mu}e^a_{\\mu}$. The antisymmetrization of the partial derivative is required by Hermiticity. Note that the action in Eq.~\\eqref{ngaplq} is  invariant under reparametrizations of the string if we choose $\\psi (x)$ to transform like a scalar, and under Lorentz symmetry if we choose $\\psi (x)$ to transform like a four-dimensional Dirac field but keeping $x$ invariant.\n\nLet us choose the Gauge or string parametrization\n\\begin{align}\n&\\xi^0=\\tau= t\\,,\\\\\n&\\xi^3=\\lambda= z\\,.\n\\end{align}\nExpanding the action in Eq.~\\eqref{gst} for small string fluctuations we arrive at\n\\begin{align}\nS_{\\rm g}=-\\sigma \\int dt dz\\left(1-\\frac{1}{2}\\pa^a\\xi^l\\pa_a\\xi^l+\\dots\\right)\\,,\\label{gstex}\n\\end{align}\nand for the case of the fermionic action in Eq.~\\eqref{ngaplq} we find\n\\begin{align}\nS_{\\rm l.q}&=\\int  dt dz\\left(\\bar{\\psi}(t,z)i\\gamma^a\\overleftrightarrow{\\partial}_a\\psi(t,z)-m_{\\rm l.q.}\\bar{\\psi}(t,z)\\psi(t,z)-\\pa^a\\xi^l\\bar{\\psi}(t,z)i\\gamma^l\\overleftrightarrow{\\partial}_a\\psi(t,z)+\\dots\\right)\\,,\\label{ngaplqexp}\n\\end{align}\nwith $l=1,2,$ and $a=0,3$.\n\nThe fermion field mode expansion is\n\\begin{align}\n\\psi(t,\\,z)=\\sum_{n=-\\infty}^\\infty\\sum_s\\frac{1}{\\sqrt{2rE_n}}\\left(u^s_+(n)a^s_n e^{ip_nz}e^{-iE_nt}+u^s_-(n)b_n^{s\\,\\dagger} e^{-ip_n z}e^{iE_nt}\\right)\\label{fme1}\n\\end{align}\nwhere $E_n=\\sqrt{p^2_n+m_{\\rm l.q.}^2}$. If we consider both periodic and antiperiodic solutions $p_n=n\\pi\/$r, $n\\in\\mathbb{Z}$. The spinors are defined as\n\\begin{align}\n&u^s_{+}(E,\\,p)=\\frac{1}{\\sqrt{E+m_{\\rm l.q.}}}\\left(\\begin{array}{c} E+m_{\\rm l.q.} \\\\ p\\sigma_3 \\end{array}\\right)\\chi_s\\,,\\\\\n&u^s_{-}(E,\\,p)=\\frac{1}{\\sqrt{E+m_{\\rm l.q.}}}\\left(\\begin{array}{c} p\\sigma_3 \\\\ E+m_{\\rm l.q.} \\end{array}\\right){\\tilde\\chi}_s,\n\\end{align}\nwith $\\chi_{+1\/2}=(1,\\,0)$, $\\chi_{-1\/2}=(0,\\,1)$ and ${\\tilde\\chi}_s=-i\\sigma_2{\\chi}_s^\\ast$. The commutation relations for the creation an annihilation operators are\n\\begin{align}\n\\{a^s_n,\\,a_{n^\\prime}^{s^\\prime\\,\\dagger}\\}&=\\delta^{sr}\\delta_{nn^\\prime}\\,,\\\\\n\\{b^s_n,\\,b_{n^\\prime}^{s^\\prime\\,\\dagger}\\}&=\\delta^{sr}\\delta_{nn^\\prime}\\,,\n\\end{align}\nall the other anticommutators vanish.\n\nThe field mode expansion in Eq.~\\eqref{fme1} contains both positive- and negative-parity modes. Since the spinors fulfill the relation $u_\\pm(E,-p)=\\pm\\gamma^0 u_\\pm(E,p)$ a convenient choice for the transformation of the creation and annihilation operators under parity is\n\\begin{align}\nPa^s_nP= a^s_{-n}\\,,\\quad Pb^s_nP=-b^s_{-n}\\label{partt}\\,.\n\\end{align}\nOne can split the field mode expansion into two components of well-defined parity with the following definitions\n\\begin{align}\n\\psi_{n\\eta_P}(t,z)=\\sum_s\\frac{1}{\\sqrt{2E_n}}&\\left[\\varphi^s_{\\eta_P\\,+}(z,\\,n)a^s_{n\\eta_P} e^{-iE_nt}+\\varphi^s_{\\eta_P\\,-}(z,\\,n)b^{s\\,\\dagger}_{n\\eta_P} e^{iE_nt}\\right]\\,,\\label{fme2}\n\\end{align}\nwith\n\\begin{align}\na^s_{n\\eta_P}&=\\frac{a^s_{n}+\\eta_P a^s_{-n}}{\\sqrt{2}}\\,,\\quad b^s_{n\\eta_P}=\\frac{b^s_{n}+\\eta_P b^s_{-n}}{\\sqrt{2}}\\,,\\\\\n\\varphi^s_{\\eta_P\\pm}(z,\\,n)&=\\frac{1}{\\sqrt{2r}}\\left(u_\\pm^s(n)e^{i\\frac{n\\pi}{r}z}+\\eta_P u_\\pm^s(-n)e^{-i\\frac{n\\pi}{r}z}\\right)\\,,\n\\end{align}\nwith $\\eta_P$ the parity eigenvalue\n\\begin{align}\nP\\psi_{n\\eta_P}(t,z)P&=\\eta_P\\gamma^0\\psi_{n\\eta_P}(t,-z)\\,.\n\\end{align}\nThe field mode expansion in Eq.~\\eqref{fme1} can be rewritten in term of the two components of well-defined parity as\n\\begin{align}\n\\psi(t,z)=\\sum^{\\infty}_{n=1}\\left(\\psi_{n+}(t,z)+\\psi_{n-}(t,z)\\right)\\,.\n\\end{align}\n\n\\subsection{Mapping}\n\nOur aim is to use the EST introduced in Sec.~\\ref{est:c} to compute the Wilson loops with operator insertions in Eqs.~\\eqref{lopotst}-\\eqref{nlopotl} which correspond to the potentials in the BOEFT. In order to do so we need a correspondence between NRQCD and EST correlators. This correspondence is defined by a mapping of NRQCD operators to the EST ones with matching symmetry properties. The symmetry transformations which leave a system of two static particles invariant form the group $D_{\\infty h}$, which is the symmetry group of a cylinder. The basic transformations are rotations around the cylinder axis, reflections across a plane including the cylinder axis and parity. The conventional notation for the representations of $D_{\\infty h}$ is $\\Lambda_{\\eta}^\\sigma$. $\\Lambda$ is the rotational quantum number, which for integer values is customarily labeled with capital Greek letters, $\\Sigma,\\,\\Pi,\\,\\Delta\\dots$ for $0\\,,1\\,,2\\dots$. The parity eigenvalue is given as the index $\\eta$ which is labeled as $g$ or $u$ for positive and negative parity, respectively. Finally, $\\sigma$ gives the sign under reflections as $+$ or $-$; however, it is only written explicitly for the $\\Sigma$ states, because for $\\Lambda > 0$ rotations around the cylinder axis mix states in this quantum number. An operator belonging to $SO(3)\\otimes P$ representation $\\kappa^{p}$ can be projected into $D_{\\infty h}$ representations: the rotational quantum number can take values corresponding to the absolute value of the projections of the spin of the operator into the heavy-quark axis $0\\leq \\Lambda\\leq|\\kappa|$ and the reflection eigenvalue corresponds to $\\sigma=\\eta(-1)^\\kappa$. To simplify, we align the heavy-quark-pair axis with the $z$-axis, i.e., $\\bm{r}=(0\\,,0\\,,z)$, set the heavy-quark positions at $z=\\pm r\/2$ and the center of mass at $\\bm{R}=\\bm{0}$.\n\nBoth Dirac and string fermions are spin-$1\/2$ fields and have the same properties under rotations and reflections. Moreover they can only be projected to $\\Lambda=1\/2$. Therefore, to find the mapping of NRQCD to the EST operators we just need to make sure that the parities coincide\n\\begin{align}\n&{\\cal Q}_{(1\/2)^+}(t,\\,\\bm{0})\\mapsto\\, P_+\\psi_{1+}(t,\\,0)\\,,\\label{map:e1}\\\\\n&{\\cal Q}_{(1\/2)^-}(t,\\,\\bm{0})\\mapsto\\, P_-\\psi_{1-}(t,\\,0)\\,,\\label{map:e2}\n\\end{align}\nwith $P_{\\pm}=(1\\pm\\gamma_0)\/2$. Now, let us focus on the mapping for the chromomagnetic field $\\bm{B}$, which can be projected into $\\Sigma^-_u$ and $\\Pi_u$ representations. Since we have chosen to align the heavy-quark-pair axis with the $z$-axis, then  $\\bm{B}^l$, $l=1,2$ and $\\bm{B}^3$ correspond to the $\\Pi_u$ and $\\Sigma^-_u$ representations, respectively. The mapping of the chromomagnetic field into string fluctuations can be found in Ref.~\\cite{PerezNadal:2008vm}. \n\\begin{eqnarray}\n\\bm{B}^l(t,z)&\\mapsto &\\Lambda'\\epsilon^{lm}\\partial_t\\partial_z\\xi^m(t,z)\\,,\\label{bos}\\\\\n\\bm{B}^3(t,z)&\\mapsto &{\\Lambda'''}\\epsilon^{lm}\\partial_t\\partial_z\\xi^l(t,z)\\partial_z\\xi^m(t,z)\\,.\\label{bos2}\n\\end{eqnarray}\nThis implies that $\\bm{B}^l$, $l=1,2$ is $\\mathcal{O}(1\/r^2)$ and $\\bm{B}^3$ is $\\mathcal{O}(1\/\\Lambda_{\\rm QCD} r^3)$. However, mappings into string fermion operators are now possible and in fact provide the leading order contribution to the potentials in Eqs.~\\eqref{nlopots1}-\\eqref{nlopotl}. This mapping is as follows:\n\\begin{align}\n\\bm{B}^l(t,\\,z)&\\mapsto\\Lambda_f\\bar{\\psi}(t,\\,z)\\frac{\\bm{\\Sigma}^l}{2}\\psi(t,\\,z)\\,\n\\label{map:e3}\\\\\n\\bm{B}^3(t,\\,z)&\\mapsto\\Lambda^{\\prime}_f\\bar{\\psi}(t,\\,z)\\frac{\\bm{\\Sigma}^3}{2}\\psi(t,\\,z)\\,,\\label{map:e4}\n\\end{align}\nwith $\\bm{\\Sigma}={\\rm diag}(\\bm{\\sigma},\\,\\bm{\\sigma})$. Note that here both $\\bm{B}^l$, $l=1,2$ and $\\bm{B}^3$ are $\\mathcal{O}(\\Lambda_{\\rm QCD}\/ r)$, and hence are more important than the corresponding bosonic operators in Eqs.~\\eqref{bos} and \\eqref{bos2}. Finally, to convert the two-dimensional spin operators in Eqs.~\\eqref{nlopots1} and \\eqref{nlopots2} into four-dimensional spin operators, we will use the following prescription \n\\begin{align}\n\\bm{S}_{1\/2}\\mapsto\\,\\frac{1}{2}\\bm{\\Sigma}\\,.\\label{map:e5}\n\\end{align}\n\n\\subsection{Long-distance potentials}\n\nUsing the mapping of NRQCD operators in the Wilson loop to EST operators defined by Eqs.~\\eqref{map:e1}-\\eqref{map:e5} we compute the potentials in Eqs.~\\eqref{lopotst}-\\eqref{nlopotl} as correlators in the EST. For example, let us apply the mapping to the Wilson loop with the insertion of just the light-quark operators in the spatial sides of the loop\n\\begin{align}\n\\langle 1\\rangle^{(1\/2)^{\\pm}}_{\\Box}\\,\\mapsto \\,P_\\pm\\langle \\psi_{1\\pm}(t\/2,\\,0)\\psi^{\\dagger}_{1\\pm}(-t\/2,\\,0)\\rangle P_\\pm&=\\frac{e^{-i(\\sigma r+E_1)t}}{rE_1}(E_1\\pm m)P_{\\pm}\n\\end{align}\nthen the static potential is just\n\\begin{align}\nV_{(1\/2)^{\\pm}}^{(0)}(\\bm{r})&=\\sigma r +E_1\\,.\n\\end{align}\nSimilarly, one can apply the mapping to compute the heavy-quark spin and angular-momentum dependent potentials\n\\begin{align}\nV^{s1}_{(1\/2)^{\\pm}}(r)&=\\frac{c_F}{3r}\\left(1\\mp\\frac{m_{\\rm l.q.}}{E_1}\\right)\\left(\\Lambda^{\\prime}_f-2\\Lambda_f\\right)\\,,\\label{ldps1}\\\\\nV^{s2}_{(1\/2)^{\\pm}}(r)&=\\frac{c_F}{r}\\left(1\\mp\\frac{m_{\\rm l.q.}}{E_1}\\right)\\left(\\Lambda^{\\prime}_f+\\Lambda_f\\right)\\,,\\label{ldps2}\\\\\nV^{l}_{(1\/2)^{\\pm}}(r)=&-\\frac{1}{2r}\\left(1\\mp\\frac{4}{\\pi^2}\\frac{m_{\\rm l.q.}}{E_1}\\right)\\Lambda_f\\,.\\label{ldpl}\n\\end{align}\n\n\\section{Doubly heavy baryon hyperfine splittings}\\label{sec:hysp}\n\\subsection{Hyperfine contributions}\\label{hysc}\n\nThe hyperfine contributions to the masses of doubly heavy baryons have been computed in Ref.~\\cite{Soto:2020pfa} for the states associated to the static energies $(1\/2)_g$ and $(1\/2)_u'$. These two static energies are interpolated by $(1\/2)^+$ and $(1\/2)^-$ light-quark operators, respectively. We summarize the quantum numbers available for the states associated to these static energies in Table~\\ref{quantumnumbers}. Since the results of this section are equivalent for both $\\kappa^p=(1\/2)^\\pm$ we will not display these labels.\n\\begin{table}[ht!]\n\\begin{tabular}{||c|c|c|c|c|c|c||}\n \\hline\\hline\n$\\kappa^p$                     & $\\Lambda_\\eta$                                  & $l$             & $\\ell$        & $s_{QQ}$ & $j$                                       & $\\eta_P$   \\\\ \\hline\n\\multirow{4}{*}{$(1\/2)^{\\pm}$} & \\multirow{4}{*}{$(1\/2)_{g\/u'}$}                 & $0$             & $1\/2$         & $1$      & $(1\/2,\\,3\/2)$                             & $\\pm$ \\\\ \n                               &                                                 & $1$             & $(1\/2,\\,3\/2)$ & $0$      & $(1\/2,\\,3\/2)$                             & $\\mp$ \\\\ \n                               &                                                 & $2$             & $(3\/2,\\,5\/2)$ & $1$      & $((1\/2,\\,3\/2,\\,5\/2)\\,,(3\/2,\\,5\/2,\\,7\/2))$ & $\\pm$ \\\\ \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t &                                                 & $3$             & $(5\/2,\\,7\/2)$ & $0$      & $(5\/2,\\,7\/2)$                             & $\\mp$ \\\\ \\hline\\hline\n\\end{tabular}\n\\caption{Quantum numbers of doubly heavy baryons associated with the $(1\/2)_{g}$ and $(1\/2)_{u}'$ static energies. The quantum numbers are as follows: $l(l+1)$ is the eigenvalue of $\\bm{L}^2_{QQ}$, $\\ell(\\ell+1)$ is the eigenvalue of $\\bm{L}^2=(\\bm{L}_{QQ}+\\bm{S}_{1\/2})^2$, $s_{QQ}(s_{QQ}+1)$ is the eigenvalue of $\\bm{S}^2_{QQ}$. Note that the Pauli exclusion principle constrains $s_{QQ}=0$ for odd $l$ and $s_{QQ}=1$ for even $l$. The total angular momentum $\\bm{J}^2=(\\bm{L}+\\bm{S}_{QQ})^2$ has eigenvalue $j(j+1)$. Finally, $\\eta_P$ stands for the parity eigenvalue. Numbers in parentheses correspond to degenerate multiplets at leading order. Notice that $\\pm$ in the parity column does not indicate degeneracy in that quantum number but correlates to the $\\pm$ parity of the light-quark operator in the first column.}\n\\label{quantumnumbers}\n\\end{table}\nLet us label the mass of the states as $M_{njl\\ell}=M^{(0)}_{nl}+M^{(1)}_{njl\\ell}+\\dots$ with $M^{(0)}_{nl}$ the mass solution of the Schr\\\"odinger equation with the static potential and $M^{(1)}_{njl\\ell}$ the hyperfine contribution. Recall that due to the Pauli principle the heavy-quark spin is $s_{QQ}=0$ for $l$ odd and $s_{QQ}=1$ for $l$ even. Let us denote the expectation values of the potentials between the radial wave functions as\n\\begin{align}\n{\\cal V}^{i}_{nl}=\\int_0^{\\infty}dr\\,r^2\\,\\psi^{nl\\,\\dagger}(r)V^{i}(r)\\psi^{nl}(r)\\,,\\quad i=s1,\\,s2,\\,l.\n\\end{align}\n\nThe hyperfine contributions for $l=0$ are given by\n\\begin{align}\nM^{(1)}_{nj0\\frac{1}{2}}=\\frac{1}{2}\\left(j(j+1)-\\frac{11}{4}\\right)\\frac{{\\cal V}^{s1}_{n0}}{m_Q}\\,,\n\\end{align}\nand the splitting is\n\\begin{align}\nM_{n\\frac{3}{2}0\\frac{1}{2}}-M_{n\\frac{1}{2}0\\frac{1}{2}}=\\frac{3}{2}\\frac{{\\cal V}^{s1}_{n0}}{m_Q}\\,.\\label{l0hfs}\n\\end{align}\nIn the case $l$ is an odd number the hyperfine contribution is as follows:\n\\begin{align}\nM^{(1)}_{njlj}=\\frac{1}{2}\\left(j(j+1)-\\frac{3}{4}-l(l+1)\\right)\\frac{{\\cal V}^{l}_{nl}}{m_Q}\\,,\n\\end{align}\nwhich for the cases $l=1,3$ leads to the following splittings\n\\begin{align}\nM_{n\\frac{3}{2}1\\frac{3}{2}}-M_{n\\frac{1}{2}1\\frac{1}{2}}&=\\frac{3}{2}\\frac{{\\cal V}^{l}_{n1}}{m_Q}\\,,\\label{l1hfs}\\\\\nM_{n\\frac{7}{2}3\\frac{7}{2}}-M_{n\\frac{5}{2}3\\frac{5}{2}}&=\\frac{7}{2}\\frac{{\\cal V}^{l}_{n3}}{m_Q}\\,\\label{l3hfs}\n\\end{align}\nFor $l=2$ the hyperfine contributions are more complicated since they depend on all three potentials in Eq.~\\eqref{nlopot} and the states $j=3\/2,5\/2$ with $\\ell=3\/2$ and $\\ell=5\/2$ are mixed. For $j=1\/2$ and $7\/2$ the contributions are\n\\begin{align}\n&M^{(1)}_{n\\frac{1}{2}2\\frac{3}{2}}=\\frac{1}{2}\\frac{{\\cal V}^{s1}_{n2}}{m_Q}-\\frac{1}{3}\\frac{{\\cal V}^{s2}_{n2}}{m_Q}-\\frac{3}{2}\\frac{{\\cal V}^{l}_{n2}}{m_Q}\\,,\\label{ml21}\\\\\n&M^{(1)}_{n\\frac{7}{2}2\\frac{5}{2}}=\\frac{1}{2}\\frac{{\\cal V}^{s1}_{n2}}{m_Q}-\\frac{2}{21}\\frac{{\\cal V}^{s2}_{n2}}{m_Q}+\\frac{{\\cal V}^{l}_{n2}}{m_Q}\\,.\\label{ml26}\n\\end{align}\nFor $j=3\/2,5\/2$ we have the mixing matrices for $\\ell=3\/2$ and $\\ell=5\/2$ states\\footnote{The off-diagonal terms were initially overlooked in Ref.~\\cite{Soto:2020pfa}. They have been included in an Erratum.} \n\\begin{align}\n&M^{(1)}_{n\\frac{3}{2}2}=\\frac{1}{m_Q}\\left(\\begin{array}{cc} \\frac{1}{5}{\\cal V}^{s1}_{n2}-\\frac{2}{15}{\\cal V}^{s2}_{n2}-\\frac{3}{2}{\\cal V}^{l}_{n2} & \\frac{3}{5}{\\cal V}^{s1}_{n2}+\\frac{1}{10}{\\cal V}^{s2}_{n2} \\\\ \\frac{3}{5}{\\cal V}^{s1}_{n2}+\\frac{1}{10}{\\cal V}^{s2}_{n2} & -\\frac{7}{10}{\\cal V}^{s1}_{n2}+\\frac{2}{15}{\\cal V}^{s2}_{n2}+{\\cal V}^{l}_{n2}\\end{array}\\right)\\,,\\\\\n&M^{(1)}_{n\\frac{5}{2}2}=\\frac{1}{m_Q}\\left(\\begin{array}{cc} -\\frac{3}{10}{\\cal V}^{s1}_{n2}+\\frac{1}{5}{\\cal V}^{s2}_{n2}-\\frac{3}{2}{\\cal V}^{l}_{n2} & \\frac{\\sqrt{14}}{5}{\\cal V}^{s1}_{n2}+\\frac{1}{15}\\sqrt{\\frac{7}{2}}{\\cal V}^{s2}_{n2} \\\\ \\frac{\\sqrt{14}}{5}{\\cal V}^{s1}_{n2}+\\frac{1}{15}\\sqrt{\\frac{7}{2}}{\\cal V}^{s2}_{n2} & -\\frac{1}{5}{\\cal V}^{s1}_{n2}+\\frac{4}{105}{\\cal V}^{s2}_{n2}+{\\cal V}^{l}_{n2}\\end{array}\\right)\\,.\n\\end{align}\nWe diagonalize to obtain the physical states\n\\begin{align}\nM^{(1)}_{n\\frac{3}{2}2\\pm}=&-\\frac{1}{4m_Q}\\left\\{{\\cal V}^{s1}_{n2}+{\\cal V}^{l}_{n2}\\pm\\frac{1}{3}\\left[81\\left({\\cal V}^{s1}_{n2}\\right)^2+4\\left({\\cal V}^{s2}_{n2}\\right)^2+225\\left({\\cal V}^{l}_{n2}\\right)^2-6{\\cal V}^{l}_{n2}\\left(27{\\cal V}^{s1}_{n2}-8{\\cal V}^{s2}_{n2}\\right)\\right]^{1\/2}\\right\\}\\,,\\\\\nM^{(1)}_{n\\frac{5}{2}2\\pm}=&-\\frac{1}{84m_Q}\\left\\{21{\\cal V}^{s1}_{n2}-10{\\cal V}^{s2}_{n2}+21{\\cal V}^{l}_{n2}\\pm\\left[3969\\left({\\cal V}^{s1}_{n2}\\right)^2+156\\left({\\cal V}^{s2}_{n2}\\right)^2+11025\\left({\\cal V}^{l}_{n2}\\right)^2+126{\\cal V}^{s1}_{n2}\\left(10{\\cal V}^{s2}_{n2}+7{\\cal V}^{l}_{n2}\\right)\\right.\\right.\\nonumber\\\\\n&\\left.\\left.-1428{\\cal V}^{s2}_{n2}{\\cal V}^{l}_{n2}\\right]^{1\/2}\\right\\}\\,.\n\\end{align}\nFor simplicity we consider the following hyperfine splittings among $l=2$ which are linear in the expectation values of the potentials\n\\begin{align}\nM_{n\\frac{5}{2}2+}+M_{n\\frac{5}{2}2-}-M_{n\\frac{3}{2}2+}-M_{n\\frac{3}{2}2-}=&\\frac{5}{21 m_Q}{\\cal V}^{s2}_{n2}\\,,\\label{l2hfs1}\\\\\nM_{n\\frac{1}{2}2\\frac{3}{2}}-\\frac{1}{2}\\left(M_{n\\frac{3}{2}2+}+M_{n\\frac{3}{2}2-}\\right)=&\\frac{1}{12m_Q}\\left(9{\\cal V}^{s1}_{n2}-4{\\cal V}^{s2}_{n2}-15{\\cal V}^{l}_{n2}\\right)\\,,\\\\\nM_{n\\frac{7}{2}2\\frac{5}{2}}-\\frac{1}{2}\\left(M_{n\\frac{3}{2}2+}+M_{n\\frac{3}{2}2-}\\right)=&\\frac{1}{m_Q}\\left(\\frac{3}{4}{\\cal V}^{s1}_{n2}-\\frac{2}{21}{\\cal V}^{s2}_{n2}+\\frac{5}{4}{\\cal V}^{l}_{n2}\\right)\\,.\\label{l2hfs3}\n\\end{align}\nThese formulas fix ${\\cal V}^{s1}_{n2}$, ${\\cal V}^{s2}_{n2}$ and ${\\cal V}^{l}_{n2}$ in terms of physical masses. Then, we have the following model-independent predictions\n\\begin{align}\nM^{(1)}_{n\\frac{3}{2}2+}-M^{(1)}_{n\\frac{3}{2}2-}=&-\\frac{1}{6m_Q}\\left[81\\left({\\cal V}^{s1}_{n2}\\right)^2+4\\left({\\cal V}^{s2}_{n2}\\right)^2+225\\left({\\cal V}^{l}_{n2}\\right)^2-6{\\cal V}^{l}_{n2}\\left(27{\\cal V}^{s1}_{n2}-8{\\cal V}^{s2}_{n2}\\right)\\right]^{1\/2}\\,,\\label{l2hfs4}\\\\\nM^{(1)}_{n\\frac{5}{2}2+}-M^{(1)}_{n\\frac{5}{2}2-}=&-\\frac{1}{42m_Q}\\left[3969\\left({\\cal V}^{s1}_{n2}\\right)^2+156\\left({\\cal V}^{s2}_{n2}\\right)^2+11025\\left({\\cal V}^{l}_{n2}\\right)^2+126{\\cal V}^{s1}_{n2}\\left(10{\\cal V}^{s2}_{n2}+7{\\cal V}^{l}_{n2}\\right)-1428{\\cal V}^{s2}_{n2}{\\cal V}^{l}_{n2}\\right]^{1\/2}\\,.\\label{l2hfs5}\n\\end{align}\n\n\\subsection{Interpolation of the full potentials}\\label{sec:intfp}\n\nWe have obtained descriptions of the potentials of the spin and angular-momentum dependent operators in the short- and long-distance regimes in Eqs.~\\eqref{sdps1}-\\eqref{sdpl} and \\eqref{ldps1}-\\eqref{ldpl}, respectively. In this section we propose an interpolation between the descriptions of the potentials in these two regions to model the potential in the intermediate distance regime $r\\sim1\/\\Lambda_{\\rm QCD}$. Using this interpolation and the wave functions obtained in Ref.~\\cite{Soto:2020pfa}, we compute the hyperfine splittings of Sec.~\\ref{hysc} in terms of the parameters of the short- and long-distance descriptions. These parameters are then determined by fitting the hyperfine splittings of lattice determinations~\\cite{Briceno:2012wt,Namekawa:2013vu,Brown:2014ena,Alexandrou:2014sha,Bali:2015lka,Padmanath:2015jea,Alexandrou:2017xwd,Lewis:2008fu,Brown:2014ena,Mohanta:2019mxo} of the double charm and bottom baryon spectrum and in the case of the short-distance parameters using heavy quark-diquark symmetry.\n\nThe interpolation we propose is constructed by summing the short- and long-distance descriptions multiplied by weight functions depending of $r$ and a new $r_0$ parameter. The weight functions are $w_s=r_0^n\/(r^n+r_0^n)$ and $w_l=r^n\/(r^n+r_0^n)$ for the short- and long-distance pieces, respectively. The sum of the weight functions is $w_s+w_l=1$ and the $r_0$ parameter determines the value of $r$ where both weights are equal. The value of the exponent $n$  is chosen as the minimal value that ensures that the product of the short- and long-distance potentials and the respective weight functions vanishes in the long- and short-distance limits, respectively. For the short-distance potentials we consider the contributions up to next-to-leading order. The resulting interpolated potentials are as follows:\n\\begin{align}\n&V^{s1\\,{\\rm int}}_{(1\/2)^\\pm}=c_F\\frac{\\left(\\Delta^{(0)}_{(1\/2)^{\\pm}}+\\Delta^{(1,0)}_{(1\/2)^{\\pm}}r^2 \\right)r^6_0+\\frac{\\left(\\Lambda^{\\prime}_f-2\\Lambda_f\\right)}{3}\\left(1\\mp\\frac{m_{\\rm l.q.}}{E_1}\\right)r^5}{r^6+r^6_0}\\,,\\label{s1int}\\\\\n&V^{s2\\,{\\rm int}}_{(1\/2)^\\pm}=c_F\\frac{\\Delta^{(1,2)}_{(1\/2)^{\\pm}}r^2 r^6_0+\\left(\\Lambda^{\\prime}_f+\\Lambda_f\\right)\\left(1\\mp\\frac{m_{\\rm l.q.}}{E_1}\\right)r^5}{r^6+r^6_0}\\,,\\\\\n&V^{l\\,{\\rm int}}_{(1\/2)^\\pm}=\\frac{1}{2}\\frac{\\left[\\Delta^{(0)}_{(1\/2)^{\\pm}}+\\left(\\Delta^{(1,0)}_{(1\/2)^{\\pm}}-\\frac{1}{3}\\Delta^{(1,2)}_{(1\/2)^{\\pm}}\\right)r^2\\right]r^6_0-\\Lambda_f\\left(1\\mp\\frac{4}{\\pi^2}\\frac{m_{\\rm l.q.}}{E_1}\\right)r^5}{r^6+r^6_0}\\,,\\label{lint}\n\\end{align}\nwith $E_1=\\sqrt{(\\pi\/r)^2+m^2_{\\rm l.q.}}$. Note that for $r_0=0$ we recover the long-distance potentials and for $r_0\\to\\infty$ we recover the short-distance potentials.\n\n\nAn accurate determination of $m_{\\rm l.q.}$ would require lattice data for the static energies at longer distances than the one currently available. Nevertheless, fitting the long-distance part of the static potential to the lattice data of Refs.~\\cite{Najjar:2009da,Najjarthesis}, we find the value \n\\begin{align}\nm_{\\rm l.q.}=0.226~{\\rm GeV}\\,,\\label{mlq}\n\\end{align}\nfor which the contribution to the potentials of the terms proportional to $m_{\\rm l.q.}$ is small.\n\nTo obtain the unknown parameters in the interpolated potentials in Eqs.~\\eqref{s1int}-\\eqref{lint} for the case $\\kappa^p=(1\/2)^+$ we minimize $\\chi^2$ function constructed as the sum of the hyperfine splittings of Sec.~\\ref{hysc} taking the masses of the doubly heavy baryons from lattice determinations. The list of contributions to the $\\chi^2$ function is as follows: For the double charm baryons $1S$ splitting in Eq.~\\eqref{l0hfs}, there are six data points corresponding to Refs.~\\cite{Briceno:2012wt,Namekawa:2013vu,Brown:2014ena,Alexandrou:2014sha,Bali:2015lka,Padmanath:2015jea,Alexandrou:2017xwd}. For the double bottom baryons $1S$ splitting, there are three data points corresponding to Refs.~\\cite{Lewis:2008fu,Brown:2014ena,Mohanta:2019mxo}. The rest are single data points for double charm baryons from Ref.~\\cite{Padmanath:2015jea} corresponding to the splittings for $2S$ and $3S$ from Eq.~\\eqref{l0hfs}, $1P$ and $2P$ from Eq.~\\eqref{l1hfs}, $1D$ and $2D$ from Eqs.~\\eqref{l2hfs1}-\\eqref{l2hfs3}, and finally, $1F$ from Eq.~\\eqref{l3hfs}. The concrete assignments of quantum numbers to the states of Ref.~\\cite{Padmanath:2015jea} that we have used are specified in Table~\\ref{latstates}.  We performed several sets of fits varying the value of $r_0$; in Table~\\ref{fits4p} we present the results with all parameters free, in Table~\\ref{fits3p} we fix $\\Delta^{(0)}_{(1\/2)^{+}}=0.122$~{GeV}$^2$ from the $B$-meson splittings in Table~\\ref{delahmm} and in Table~\\ref{fits2p} we fix $\\Delta^{(0)}_{(1\/2)^{+}}=0.122$~{GeV}$^2$ and set $\\Delta^{(1,0)}_{(1\/2)^{+}}=\\Delta^{(1,2)}_{(1\/2)^{+}}=0$~{GeV}$^4$.\n\n\\begin{table}\n\\begin{tabular}{||c|c|c|c|c||} \\hline \\hline\n                $l$     & $n$                  & $j$      & $M-M_{\\eta_c}[{\\rm GeV}]$ \\\\ \\hline                      \n\\multirow{6}{*}{$0$}    & \\multirow{2}{*}{$1$} & $1\/2$    & $0.6532(80)$   \\\\ \\cline{3-4}\n                        &                      & $3\/2$    & $0.7474(88)$   \\\\ \\cline{2-4}\n                        & \\multirow{2}{*}{$2$} & $1\/2$    & $1.3163(216)$  \\\\ \\cline{3-4}\n                        &                      & $3\/2$    & $1.3297(332)$  \\\\ \\cline{2-4}\n                        & \\multirow{2}{*}{$3$} & $1\/2$    & $1.5427(142)$  \\\\ \\cline{3-4}\n                        &                      & $3\/2$    & $1.5435(291)$  \\\\ \\hline\n\\multirow{4}{*}{$1$}  & \\multirow{2}{*}{$1$} & $1\/2$    & $1.0243(114)$  \\\\ \\cline{3-4}\n                        &                      & $3\/2$    & $1.0733(113)$  \\\\ \\cline{2-4}\n                        & \\multirow{2}{*}{$2$} & $1\/2$    & $1.5829(296)$  \\\\ \\cline{3-4}\n                        &                      & $3\/2$    & $1.6315(353)$  \\\\ \\hline          \n\\multirow{12}{*}{$2$} & \\multirow{7}{*}{$1$} & $1\/2$    & $1.3114(213)$  \\\\ \\cline{3-4}\n                        &                      & $3\/2\\,+$ & $1.2653(232)$  \\\\ \\cline{3-4}\n &                      & $3\/2\\,-$ & $1.3697(131)$  \\\\ \\cline{3-4}\n                        &                      & $5\/2\\,+$ & $1.3075(130)$  \\\\ \\cline{3-4}\n                        &                      & $5\/2\\,-$ & $1.3542(141)$  \\\\ \\cline{3-4}\n &                      & $7\/2$    & $1.3715(97)$   \\\\ \\cline{2-4}\n                        & \\multirow{7}{*}{$2$} & $1\/2$    & $1.5044(181)$  \\\\ \\cline{3-4}\n                        &                      & $3\/2\\,+$ & $1.4243(296)$  \\\\ \\cline{3-4}\n &                      & $3\/2\\,-$ & $1.5331(222)$  \\\\ \\cline{3-4}\n                        &                      & $5\/2\\,+$ & $1.5017(193)$  \\\\ \\cline{3-4}\n                        &                      & $5\/2\\,-$ & $1.5127(157)$  \\\\ \\cline{3-4}\n &                      & $7\/2$    & $1.5366(154)$  \\\\ \\hline\n\\multirow{2}{*}{$3$}    & \\multirow{2}{*}{$1$} & $5\/2$    & $1.5502(221)$  \\\\ \\cline{3-4}\n                        &                      & $7\/2$    & $1.5618(678)$  \\\\ \\hline\\hline\n\\end{tabular}\n\\caption{Assignments of quantum numbers of the lattice states of Ref.~\\cite{Padmanath:2015jea} used in the fits of Sec.~\\ref{sec:intfp}.}\n\\label{latstates}\n\\end{table}\n\n\n\\begin{table}\n\\begin{tabular}{||ccccccc||}\\hline\\hline\n $r_0~[{\\rm fm}]$ & $\\Delta^{(0)}_{(1\/2)^{+}}~[{\\rm GeV}^2]$  & $\\Delta^{(1,0)}_{(1\/2)^{+}}~[{\\rm GeV}^4]$  & $\\Delta^{(1,2)}_{(1\/2)^{+}}~[{\\rm GeV}^4]$ & $\\Lambda_f~[{\\rm GeV}]$ & $\\Lambda_f'~[{\\rm GeV}]$ & $\\chi^2_{\\rm d.o.f}$\\\\\\hline\n$0.0$    & n\/a          & n\/a         & n\/a           & $-0.341(8)$  & $-0.268(16)$  & $0.62$ \\\\   \n$0.1$    & $-3.13(12)$  & $15.17(37)$ & $19(77)$      & $-0.231(10)$ & $-0.282(19)$  & $0.67$ \\\\\n$0.2$    & $-0.076(22)$ & $0.514(22)$ & $0.35(1.76)$  & $-0.196(13)$ & $-0.274(23)$  & $0.64$ \\\\\n$0.3$    & $0.135(10)$  & $0.047(5)$  & $-0.045(203)$ & $-0.169(18)$ & $-0.264(32)$  & $0.63$ \\\\\n$0.4$    & $0.163(6)$   & $-0.006(2)$ & $-0.041(64)$  & $-0.154(27)$ & $-0.272(45)$  & $0.63$ \\\\\n$0.5$    & $0.165(5)$   & $-0.016(1)$ & $-0.023(26)$  & $-0.176(43)$ & $-0.322(64)$  & $0.64$ \\\\\n$0.6$    & $0.159(4)$   & $-0.016(1)$ & $-0.012(14)$  & $-0.256(67)$ & $-0.427(90)$  & $0.66$ \\\\\n$\\infty$ & $0.086(3)$   & $-0.002(1)$ & $-0.002(2)$   & n\/a          & n\/a           & $0.94$ \\\\\\hline\\hline\n\\end{tabular}\n\\caption{Global fit of $\\kappa^p=(1\/2)^+$ $l=0,1,2,3$ multiplets hyperfine splittings for all the lattice data available for various values of $r_0$.}\n\\label{fits4p}\n\\end{table}\n\n\\begin{table}\n\\begin{tabular}{||cccccc||}\\hline\\hline\n$r_0~[{\\rm fm}]$  & $\\Delta^{(1,0)}_{(1\/2)^{+}}~[{\\rm GeV}^4]$ & $\\Delta^{(1,2)}_{(1\/2)^{+}}~[{\\rm GeV}^4]$ & $\\Lambda_f~[{\\rm GeV}]$ & $\\Lambda_f'~[{\\rm GeV}]$ & $\\chi^2_{\\rm d.o.f}$\\\\\\hline\n$0.1$    & $1.89(37)$    & $-8.5(77.2)$  & $-0.308(10)$ & $-0.267(19)$ & $0.66$ \\\\\n$0.2$    & $0.231(22)$   & $0.39(1.72)$  & $-0.249(12)$ & $-0.283(23)$ & $0.62$ \\\\\n$0.3$    & $0.056(5)$    & $-0.055(226)$ & $-0.158(18)$ & $-0.258(31)$ & $0.59$ \\\\\n$0.4$    & $0.013(2)$    & $-0.061(63)$  & $-0.086(27)$ & $-0.223(44)$ & $0.61$ \\\\\n$0.5$    & $-0.0006(14)$ & $-0.036(26)$  & $-0.048(43)$ & $-0.216(64)$ & $0.64$ \\\\\n$0.6$    & $-0.005(1)$   & $-0.019(14)$  & $-0.061(67)$ & $-0.262(91)$ & $0.66$ \\\\ \n$\\infty$ & $-0.0054(5)$  & $-0.0089(29)$ & n\/a          & n\/a          & $2.67$ \\\\ \\hline\\hline\n\\end{tabular}\n\\caption{Global fit of $\\kappa^p=(1\/2)^+$ $l=0,1,2,3$ multiplets hyperfine splittings for all the lattice data available for various values of $r_0$ with $\\Delta^{(0)}_{(1\/2)^{+}}=0.122$~{GeV}$^2$ from the $B$-meson splittings in Table~\\ref{delahmm}.}\n\\label{fits3p}\n\\end{table}\n\n\n\\begin{table}\n\\begin{tabular}{||cccc||}\\hline\\hline\n$r_0~[{\\rm fm}]$ & $\\Lambda_f~[{\\rm GeV}]$ & $\\Lambda_f'~[{\\rm GeV}]$ & $\\chi^2_{\\rm d.o.f}$\\\\\\hline\n$0.1$    & $-0.355(10)$ & $-0.265(19)$  & $0.66$ \\\\\n$0.2$    & $-0.368(13)$ & $-0.264(25)$  & $0.72$ \\\\\n$0.3$    & $-0.348(19)$ & $-0.270(33)$  & $0.69$ \\\\\n$0.4$    & $-0.266(27)$ & $-0.286(44)$  & $0.60$ \\\\\n$0.5$    & $-0.085(41)$ & $-0.314(61)$  & $0.58$ \\\\\n$0.6$    & $ 0.224(75)$ & $-0.353(102)$ & $0.83$ \\\\  \\hline\\hline\n\\end{tabular}\n\\caption{Global fit of $\\kappa^p=(1\/2)^+$ $l=0,1,2,3$ multiplets hyperfine splittings for all the lattice data available for various values of $r_0$ with $\\Delta^{(0)}_{(1\/2)^{+}}=0.122$~{GeV}$^2$ from the $B$-meson splittings in Table~\\ref{delahmm} and $\\Delta^{(1,0)}_{(1\/2)^{+}}=\\Delta^{(1,2)}_{(1\/2)^{+}}=0$.}\n\\label{fits2p}\n\\end{table}\n\nSeveral conclusions can be extracted from the fits. First of all, when we restrict the fit to either the short-distance form of potential ($r_0=\\infty$) or the long-distance form of it ($r_0=0$), we see from Table \\ref{fits4p} that the latter produces a much better fit than the former. This indicates both that the long distance form is important and that the EST provides a good description of it. We observe that the value of $\\Delta^{(1,2)}_{(1\/2)^{+}}$ changes significantly, carries large uncertainty, and in the best fits it is compatible with $0$. In the case of $\\Delta^{(1,0)}_{(1\/2)^{+}}$ its value also shows variation, however it is significantly different from zero. Nevertheless, the inclusion of the next-to-leading order terms in the short-distance potentials does not improve the overall quality of the fits. Therefore, it seems that with the current lattice data it is not possible to constrain these next-to-leading order terms in the multipole expansion. The values of $\\Lambda_f$ and $\\Lambda_f'$ stay consistent across the different sets of fits, with $\\Lambda_f'$ being very stable while $\\Lambda_f$ decreasing in absolute value as $r_0$ gets larger and even changing sign. Our preferred fit is the one with minimal $\\chi^2_{\\rm d.o.f}$ in Table~\\ref{fits2p} corresponding to $r_0=0.5$~fm. In Fig.~\\ref{plot_pot} we plot the interpolated expressions of the potentials in Eqs.~\\eqref{s1int}-\\eqref{lint} for the parameter set in the entry for $r_0=0.5$~fm in Table~\\ref{fits2p} and  $r_0=0.3$~fm in Table~\\ref{fits4p}. In the case of the potentials for $\\kappa^p=(1\/2)^-$, we plot the potentials with $\\Delta^{(0)}_{(1\/2)^{-}}=0.075$~GeV$^2$ from the neutral $D$-meson entry in Table~\\ref{delahmm}, $\\Delta^{(1,0)}_{(1\/2)^{-}}=\\Delta^{(1,2)}_{(1\/2)^{-}}=0$~GeV$^4$ and the values of $\\Lambda_f$ and $\\Lambda_f'$ from the entries for $r_0=0.5$~fm in Table~\\ref{fits2p} and $r_0=0.3$~fm in Table~\\ref{fits4p}. Although in some cases the potentials in Fig.~\\ref{plot_pot} show significant variation depending on the parameter set used, we will show in the following section that this is not the case for the values of the hyperfine splittings.\n\n\\begin{figure}\n\\includegraphics[width=0.4\\linewidth]{vs1.pdf}\\includegraphics[width=0.4\\linewidth]{vs1m.pdf} \\\\\n\\includegraphics[width=0.4\\linewidth]{vs2.pdf}\\includegraphics[width=0.4\\linewidth]{vs2m.pdf} \\\\\n\\includegraphics[width=0.4\\linewidth]{vl.pdf}\\includegraphics[width=0.4\\linewidth]{vlm.pdf} \\\\\n\\caption{Plot of the potentials in Eqs.~\\eqref{s1int}-\\eqref{lint} for the values of the parameters of $r_0=0.5$~fm in Table~\\ref{fits2p} and  $r_0=0.3$~fm in Table~\\ref{fits4p}. In the case of $\\kappa^p=(1\/2)^-$ we take $\\Delta^{(0)}_{(1\/2)^{-}}=0.075$~GeV$^2$ and $\\Delta^{(1,2)}_{(1\/2)^{-}}=0$~GeV$^4$ and the values of $\\Lambda_f$ and $\\Lambda_f'$ indicated in the legend. In the potentials $V^{s1}$ and $V^{s2}$ we use the two-loop, RG improved expression of $c_F=c_F(1~{\\rm GeV},\\,m_c)$.}\n\\label{plot_pot}\n\\end{figure}\n\n\\subsection{Doubly heavy baryon spectra}\\label{sec:dhbs}\n\nNow we compute the spectrum of double charm and bottom baryons including the hyperfine contributions using the interpolated potentials in Eqs.~\\eqref{s1int}-\\eqref{lint}. For the states associated to the $(1\/2)_g$ static energy, we take values of the parameters from the entry $r_0=0.5$~fm in Table~\\ref{fits2p} and for comparison the entry $r_0=0.3$~fm in Table~\\ref{fits4p}. The results can be found in Tables~\\ref{hft12gc} and \\ref{hft12gb} for double charm and double bottom baryons, respectively. For the states associated to the $(1\/2)_u'$ static energy we set $\\Delta^{(0)}_{(1\/2)^{-}}=0.075$~GeV$^2$ and $\\Delta^{(1,0)}_{(1\/2)^{-}}=\\Delta^{(1,2)}_{(1\/2)^{-}}=0$~GeV$^4$ and take the values of $\\Lambda_f$ and $\\Lambda_f'$ from the $r_0=0.5$~fm entry in Table~\\ref{fits2p} and for comparison the entry $r_0=0.3$~fm in Table~\\ref{fits4p}. The results can be found in Tables~\\ref{hft12upc} and \\ref{hft12upb} for double charm and bottom baryons respectively. The results for the spectra for the two sets of parameters are very close.\n\n\\begin{table}\n\\begin{tabular}{||c|c|c|c|c|c|c|c||} \\hline \\hline\n\\multirow{2}{*}{$l$}    & \\multirow{2}{*}{$n$} & \\multirow{2}{*}{$M^{(0)}$} & \\multirow{2}{*}{$j$} & \\multicolumn{2}{c|}{$r_0=0.5$~fm\\,Table~\\ref{fits2p}} & \\multicolumn{2}{c||}{$r_0=0.3$~fm\\,Table~\\ref{fits4p}}  \\\\ \\cline{5-8}\n                        &                      &                          &       & $M^{(1)}$    & $M$     & $M^{(1)}$    & $M$     \\\\ \\hline \n\\multirow{7}{*}{$0$}    & \\multirow{2}{*}{$1$} & \\multirow{2}{*}{$3.712$} & $1\/2$ & $-0.059(2)$  & $3.653$ & $-0.058(5)$  & $3.654$ \\\\ \\cline{4-8}\n                        &                      &                          & $3\/2$ & $0.029(1)$   & $3.741$ & $0.029(2)$   & $3.741$ \\\\ \\cline{2-8}\n                        & \\multirow{2}{*}{$2$} & \\multirow{2}{*}{$4.286$} & $1\/2$ & $-0.020(2)$  & $4.266$ & $-0.029(3)$  & $4.257$ \\\\ \\cline{4-8}\n                        &                      &                          & $3\/2$ & $0.010(1)$   & $4.296$ & $0.015(1)$   & $4.301$ \\\\ \\cline{2-8}\n                        & \\multirow{2}{*}{$3$} & \\multirow{2}{*}{$4.748$} & $1\/2$ & $-0.013(2)$  & $4.735$ & $-0.020(2)$  & $4.728$ \\\\ \\cline{4-8}\n                        &                      &                          & $3\/2$ & $0.007(1)$   & $4.755$ & $0.010(1)$   & $4.758$ \\\\ \\hline\n\\multirow{4}{*}{$1$}  & \\multirow{2}{*}{$1$} & \\multirow{2}{*}{$4.062$}   & $1\/2$ & $-0.034(4)$  & $4.028$ & $-0.035(8)$  & $4.027$ \\\\ \\cline{4-8}\n                        &                      &                          & $3\/2$ & $0.017(2)$   & $4.079$ & $0.017(4)$   & $4.079$ \\\\ \\cline{2-8}\n                        & \\multirow{2}{*}{$2$} & \\multirow{2}{*}{$4.552$} & $1\/2$ & $-0.024(3)$  & $4.528$ & $-0.026(6)$  & $4.526$ \\\\ \\cline{4-8}\n                        &                      &                          & $3\/2$ & $0.012(1)$   & $4.564$ & $0.013(3)$   & $4.565$ \\\\ \\hline               \n\\multirow{12}{*}{$2$} & \\multirow{7}{*}{$1$} & \\multirow{7}{*}{$4.353$}   & $1\/2$ & $-0.020(7)$  & $4.333$ & $-0.009(8)$  & $4.344$ \\\\ \\cline{4-8}\n                        &                      &                          & $3\/2$ & $-0.032(6)$  & $4.321$ & $-0.026(6)$  & $4.327$ \\\\ \\cline{4-8}\n\t\t\t\t\t\t\t\t\t\t\t  &                      &                          & $3\/2$ & $0.015(4)$   & $4.368$ & $0.009(5)$   & $4.362$ \\\\ \\cline{4-8}\n                        &                      &                          & $5\/2$ & $-0.052(6)$  & $4.301$ & $-0.053(5)$  & $4.300$ \\\\ \\cline{4-8}\n                        &                      &                          & $5\/2$ & $0.023(3)$   & $4.376$ & $0.020(4)$   & $4.373$ \\\\ \\cline{4-8}\n\t\t\t\t\t\t\t\t\t\t\t  &                      &                          & $7\/2$ & $0.035(4)$   & $4.388$ & $0.035(4)$   & $4.388$ \\\\ \\cline{2-8}\n                        & \\multirow{7}{*}{$2$} & \\multirow{7}{*}{$4.794$} & $1\/2$ & $-0.017(5)$  & $4.777$ & $-0.008(8)$  & $4.786$ \\\\ \\cline{4-8}\n                        &                      &                          & $3\/2$ & $-0.026(4)$  & $4.768$ & $-0.022(6)$  & $4.772$ \\\\ \\cline{4-8}\n\t\t\t\t\t\t\t\t\t\t\t  &                      &                          & $3\/2$ & $0.011(3)$   & $4.805$ & $0.007(4)$   & $4.801$ \\\\ \\cline{4-8}\n                        &                      &                          & $5\/2$ & $-0.042(4)$  & $4.752$ & $-0.044(4)$  & $4.750$ \\\\ \\cline{4-8}\n                        &                      &                          & $5\/2$ & $0.019(3)$   & $4.813$ & $0.017(4)$   & $4.811$ \\\\ \\cline{4-8}\n\t\t\t\t\t\t\t\t\t\t\t  &                      &                          & $7\/2$ & $0.029(3)$   & $4.823$ & $0.030(4)$   & $4.824$ \\\\ \\hline\n\\multirow{2}{*}{$3$}    & \\multirow{2}{*}{$1$} & \\multirow{2}{*}{$4.612$} & $5\/2$ & $-0.043(8)$  & $4.569$ & $-0.037(5)$  & $4.575$ \\\\ \\cline{4-8}\n                        &                      &                          & $7\/2$ & $0.032(6)$   & $4.644$ & $0.028(4)$   & $4.640$ \\\\ \\hline\\hline\n\\end{tabular}\n\\caption{Hyperfine contributions to the double charm baryons for the $(1\/2)_g$ static energy for two sets of parameters of the hyperfine potentials. All masses in GeV units.}\n\\label{hft12gc}\n\\end{table}\n\n\\begin{table}\n\\begin{tabular}{||c|c|c|c|c|c|c|c||} \\hline \\hline\n\\multirow{2}{*}{$l$}    & \\multirow{2}{*}{$n$} & \\multirow{2}{*}{$M^{(0)}$} & \\multirow{2}{*}{$j$} & \\multicolumn{2}{c|}{$r_0=0.5$~fm\\,Table~\\ref{fits2p}} & \\multicolumn{2}{c||}{$r_0=0.3$~fm\\,Table~\\ref{fits4p}}  \\\\ \\cline{5-8}\n                        &                      &                            &       & $M^{(1)}$ & $M$      & $M^{(1)}$ & $M$ \\\\ \\hline \n\\multirow{8}{*}{$0$}  & \\multirow{2}{*}{$1$} & \\multirow{2}{*}{$10.140$}    & $1\/2$ & $-0.020(1)$  & $10.120$ & $-0.023(1)$  & $10.117$ \\\\ \\cline{4-8}\n                        &                      &                            & $3\/2$ & $0.010(0)$   & $10.150$ & $0.011(1)$   & $10.151$ \\\\ \\cline{2-8}\n                        & \\multirow{2}{*}{$2$} & \\multirow{2}{*}{$10.542$}  & $1\/2$ & $-0.009(1)$  & $10.533$ & $-0.009(1)$  & $10.533$ \\\\ \\cline{4-8}\n                        &                      &                            & $3\/2$ & $0.004(0)$   & $10.546$ & $0.005(0)$   & $10.547$ \\\\ \\cline{2-8}\n                        & \\multirow{2}{*}{$3$} & \\multirow{2}{*}{$10.856$}  & $1\/2$ & $-0.006(1)$  & $10.850$ & $-0.006(0)$  & $10.850$ \\\\ \\cline{4-8}\n                        &                      &                            & $3\/2$ & $0.003(0)$   & $10.859$ & $0.003(0)$   & $10.859$ \\\\ \\cline{2-8}\n\t\t\t\t\t\t\t\t\t\t    & \\multirow{2}{*}{$4$} & \\multirow{2}{*}{$11.131$}  & $1\/2$ & $-0.004(0)$  & $11.127$ & $-0.005(1)$  & $11.126$ \\\\ \\cline{4-8}\n                        &                      &                            & $3\/2$ & $0.002(0)$   & $11.133$ & $0.003(0)$   & $11.134$ \\\\ \\hline\n\\multirow{6}{*}{$1$}  & \\multirow{2}{*}{$1$} & \\multirow{2}{*}{$10.398$}    & $1\/2$ & $-0.012(1)$  & $10.386$ & $-0.016(5)$  & $10.382$ \\\\ \\cline{4-8}\n                        &                      &                            & $3\/2$ & $0.006(0)$   & $10.404$ & $0.008(3)$   & $10.406$ \\\\ \\cline{2-8}\n                        & \\multirow{2}{*}{$2$} & \\multirow{2}{*}{$10.731$}  & $1\/2$ & $-0.010(1)$  & $10.721$ & $-0.011(3)$  & $10.720$ \\\\ \\cline{4-8}\n                        &                      &                            & $3\/2$ & $0.005(1)$   & $10.736$ & $0.006(1)$   & $10.737$ \\\\ \\cline{2-8}      \n\t\t\t\t\t\t\t\t\t\t    & \\multirow{2}{*}{$3$} & \\multirow{2}{*}{$11.016$}  & $1\/2$ & $-0.008(1)$  & $11.008$ & $-0.009(2)$  & $11.007$ \\\\ \\cline{4-8}\n                        &                      &                            & $3\/2$ & $0.004(0)$   & $11.020$ & $0.004(1)$   & $11.020$ \\\\ \\hline      \n\\multirow{18}{*}{$2$} & \\multirow{7}{*}{$1$} & \\multirow{7}{*}{$10.600$}    & $1\/2$ & $-0.009(2)$  & $10.591$ & $-0.007(7)$  & $10.593$ \\\\ \\cline{4-8}\n                        &                      &                            & $3\/2$ & $-0.015(2)$  & $10.585$ & $-0.014(5)$  & $10.586$ \\\\ \\cline{4-8}\n\t\t\t\t\t\t\t\t\t\t\t  &                      &                            & $3\/2$ & $0.005(1)$   & $10.605$ & $0.004(4)$   & $10.604$ \\\\ \\cline{4-8}\n                        &                      &                            & $5\/2$ & $-0.023(2)$  & $10.577$ & $-0.026(4)$  & $10.574$ \\\\ \\cline{4-8}\n                        &                      &                            & $5\/2$ & $0.010(1)$   & $10.610$ & $0.010(3)$   & $10.610$ \\\\ \\cline{4-8}\n\t\t\t\t\t\t\t\t\t\t\t  &                      &                            & $7\/2$ & $0.017(1)$   & $10.617$ & $0.018(3)$   & $10.618$ \\\\ \\cline{2-8}\n                        & \\multirow{7}{*}{$2$} & \\multirow{7}{*}{$10.897$}  & $1\/2$ & $-0.007(2)$  & $10.890$ & $-0.006(5)$  & $10.891$ \\\\ \\cline{4-8}\n                        &                      &                            & $3\/2$ & $-0.011(1)$  & $10.886$ & $-0.011(4)$  & $10.886$ \\\\ \\cline{4-8}\n\t\t\t\t\t\t\t\t\t\t\t  &                      &                            & $3\/2$ & $0.004(1)$   & $10.901$ & $0.003(3)$   & $10.900$ \\\\ \\cline{4-8}\n                        &                      &                            & $5\/2$ & $-0.017(1)$  & $10.880$ & $-0.020(3)$  & $10.877$ \\\\ \\cline{4-8}\n                        &                      &                            & $5\/2$ & $0.008(1)$   & $10.905$ & $0.008(2)$   & $10.905$ \\\\ \\cline{4-8}\n\t\t\t\t\t\t\t\t\t\t\t  &                      &                            & $7\/2$ & $0.012(1)$   & $10.909$ & $0.015(3)$   & $10.912$ \\\\ \\cline{2-8}\n\t\t\t\t\t\t\t\t\t\t\t  & \\multirow{7}{*}{$3$} & \\multirow{7}{*}{$11.162$}  & $1\/2$ & $-0.006(1)$  & $11.156$ & $-0.005(4)$  & $11.157$ \\\\ \\cline{4-8}\n                        &                      &                            & $3\/2$ & $-0.009(1)$  & $11.153$ & $-0.010(3)$  & $11.152$ \\\\ \\cline{4-8}\n\t\t\t\t\t\t\t\t\t\t\t  &                      &                            & $3\/2$ & $0.004(1)$   & $11.166$ & $0.003(2)$   & $11.165$ \\\\ \\cline{4-8}\n                        &                      &                            & $5\/2$ & $-0.014(1)$  & $11.148$ & $-0.017(2)$  & $11.145$ \\\\ \\cline{4-8}\n                        &                      &                            & $5\/2$ & $0.007(1)$   & $11.169$ & $0.007(2)$   & $11.169$ \\\\ \\cline{4-8}\n\t\t\t\t\t\t\t\t\t\t\t  &                      &                            & $7\/2$ & $0.010(1)$   & $11.172$ & $0.012(2)$   & $11.174$ \\\\ \\hline\n\\multirow{4}{*}{$3$}  & \\multirow{2}{*}{$1$} & \\multirow{2}{*}{$10.777$}    & $5\/2$ & $-0.020(3)$  & $10.757$ & $-0.019(4)$  & $10.758$ \\\\ \\cline{4-8}\n                        &                      &                            & $7\/2$ & $0.015(2)$   & $10.792$ & $0.014(3)$   & $10.791$ \\\\ \\cline{2-8}\n\t\t\t\t\t\t\t\t\t\t    & \\multirow{2}{*}{$2$} & \\multirow{2}{*}{$11.051$}  & $5\/2$ & $-0.016(2)$  & $11.035$ & $-0.017(3)$  & $11.034$ \\\\ \\cline{4-8}\n                        &                      &                            & $7\/2$ & $0.012(2)$   & $11.063$ & $0.012(2)$   & $11.063$ \\\\ \\hline\\hline\n\\end{tabular}\n\\caption{Hyperfine contributions to the double bottom baryons for the $(1\/2)_g$ static energy for two sets of parameters of the hyperfine potentials. All masses in GeV units.}\n\\label{hft12gb}\n\\end{table}\n\n\n\\begin{table}\n\\begin{tabular}{||c|c|c|c|c|c|c|c||} \\hline \\hline\n\\multirow{2}{*}{$l$}   & \\multirow{2}{*}{$n$} & \\multirow{2}{*}{$M^{(0)}$} & \\multirow{2}{*}{$j$} & \\multicolumn{2}{c|}{$r_0=0.5$~fm\\,table~\\ref{fits2p}} &  \\multicolumn{2}{c||}{$r_0=0.3$~fm\\,table~\\ref{fits4p}}  \\\\ \\cline{5-8}\n                       &                      &                            &       & $M^{(1)}$ & $M$     & $M^{(1)}$ & $M$     \\\\ \\hline \n  \\multirow{4}{*}{$0$} & \\multirow{2}{*}{$1$} & \\multirow{2}{*}{$4.095$}   & $1\/2$ & $-0.033(9)$  & $4.062$ & $-0.025(6)$  & $4.070$ \\\\ \\cline{4-8}\n                       &                      &                            & $3\/2$ & $0.016(4)$   & $4.111$ & $0.012(3)$   & $4.107$ \\\\ \\cline{2-8}\n                       & \\multirow{2}{*}{$2$} & \\multirow{2}{*}{$4.667$}   & $1\/2$ & $-0.07(5)$   & $4.660$ & $-0.015(4)$  & $4.652$ \\\\ \\cline{4-8}\n                       &                      &                            & $3\/2$ & $0.003(2)$   & $4.670$ & $0.008(2)$   & $4.675$ \\\\ \\hline\n\\multirow{2}{*}{$1$}   & \\multirow{2}{*}{$1$} & \\multirow{2}{*}{$4.443$}   & $1\/2$ & $-0.033(5)$  & $4.410$ & $-0.033(4)$  & $4.410$ \\\\ \\cline{4-8}\n                       &                      &                            & $3\/2$ & $0.016(3)$   & $4.459$ & $0.016(2)$   & $4.459$ \\\\ \\hline               \n\\multirow{7}{*}{$2$}   & \\multirow{7}{*}{$1$} & \\multirow{7}{*}{$4.732$}   & $1\/2$ & $-0.017(9)$  & $4.715$ & $0.000(6)$   & $4.732$ \\\\ \\cline{4-8}\n                       &                      &                            & $3\/2$ & $-0.035(7)$  & $4.697$ & $-0.025(4)$  & $4.707$ \\\\ \\cline{4-8}\n\t\t\t\t\t\t\t\t\t\t\t &                      &                            & $3\/2$ & $ 0.022(5)$  & $4.754$ & $0.008(3)$   & $4.740$ \\\\ \\cline{4-8}\n                       &                      &                            & $5\/2$ & $-0.063(7)$  & $4.669$ & $-0.066(5)$  & $4.666$ \\\\ \\cline{4-8}\n                       &                      &                            & $5\/2$ & $0.029(4)$   & $4.761$ & $0.022(3)$   & $4.754$ \\\\ \\cline{4-8}\n\t\t\t\t\t\t\t\t\t\t\t &                      &                            & $7\/2$ & $0.037(6)$   & $4.769$ & $0.041(4)$   & $4.773$ \\\\ \\hline\\hline\n\\end{tabular}\n\\caption{Hyperfine contributions to the double charm baryons for the $(1\/2)_u'$ static energy for two sets of parameters $\\Lambda_f$, $\\Lambda_f'$ of the hyperfine potentials ($\\Delta^{(0)}_{(1\/2)^{-}}=0.075$~GeV$^2$, $\\Delta^{(1,0)}_{(1\/2)^{-}}=\\Delta^{(1,2)}_{(1\/2)^{-}}=0$~GeV$^4$). All masses in GeV units.}\n\\label{hft12upc}\n\\end{table}\n\n\\begin{table}\n\\begin{tabular}{||c|c|c|c|c|c|c|c||} \\hline \\hline\n\\multirow{2}{*}{$l$}   & \\multirow{2}{*}{$n$} & \\multirow{2}{*}{$M^{(0)}$} & \\multirow{2}{*}{$j$} & \\multicolumn{2}{c|}{$r_0=0.5$~fm\\,Table~\\ref{fits2p}} &  \\multicolumn{2}{c||}{$r_0=0.3$~fm\\,Table~\\ref{fits4p}}  \\\\ \\cline{5-8}\n                       &                      &                            &       & $M^{(1)}$ & $M$      & $M^{(1)}$ & $M$      \\\\ \\hline \n  \\multirow{4}{*}{$0$} & \\multirow{2}{*}{$1$} & \\multirow{2}{*}{$10.527$}  & $1\/2$ & $-0.012(2)$  & $10.515$ & $-0.010(2)$  & $10.517$ \\\\ \\cline{4-8}\n                       &                      &                            & $3\/2$ & $0.006(1)$   & $10.533$ & $0.005(1)$   & $10.532$ \\\\ \\cline{2-8}\n                       & \\multirow{2}{*}{$2$} & \\multirow{2}{*}{$10.924$}  & $1\/2$ & $-0.004(1)$  & $10.920$ & $-0.005(1)$  & $10.919$ \\\\ \\cline{4-8}\n                       &                      &                            & $3\/2$ & $0.002(1)$   & $10.926$ & $0.002(1)$   & $10.926$ \\\\ \\hline\n  \\multirow{4}{*}{$1$} & \\multirow{2}{*}{$1$} & \\multirow{2}{*}{$10.781$}  & $1\/2$ & $-0.010(1)$  & $10.771$ & $-0.012(1)$  & $10.769$ \\\\ \\cline{4-8}\n                       &                      &                            & $3\/2$ & $0.005(1)$   & $10.786$ & $0.006(1)$   & $10.787$ \\\\ \\cline{2-8}\n                       & \\multirow{2}{*}{$2$} & \\multirow{2}{*}{$11.112$}  & $1\/2$ & $-0.009(1)$  & $11.103$ & $-0.009(1)$  & $11.103$ \\\\ \\cline{4-8}\n                       &                      &                            & $3\/2$ & $0.005(1)$   & $11.117$ & $0.005(0)$   & $11.117$ \\\\ \\hline      \n\\multirow{7}{*}{$2$}   & \\multirow{7}{*}{$1$} & \\multirow{7}{*}{$10.981$}  & $1\/2$ & $-0.008(2)$  & $10.973$ & $-0.004(2)$  & $10.977$ \\\\ \\cline{4-8}\n                       &                      &                            & $3\/2$ & $-0.012(2)$  & $10.969$ & $-0.011(2)$  & $10.970$ \\\\ \\cline{4-8}\n\t\t\t\t\t\t\t\t\t\t\t &                      &                            & $3\/2$ & $0.005(1)$   & $10.986$ & $0.004(1)$   & $10.985$ \\\\ \\cline{4-8}\n                       &                      &                            & $5\/2$ & $-0.020(2)$  & $10.961$ & $-0.024(2)$  & $10.957$ \\\\ \\cline{4-8}\n                       &                      &                            & $5\/2$ & $0.009(1)$   & $10.990$ & $0.009(1)$   & $10.990$ \\\\ \\cline{4-8}\n\t\t\t\t\t\t\t\t\t\t\t &                      &                            & $7\/2$ & $0.014(2)$   & $10.995$ & $0.016(1)$   & $10.997$ \\\\ \\hline\n\\multirow{2}{*}{$3$}   & \\multirow{2}{*}{$1$} & \\multirow{2}{*}{$11.157$}  & $5\/2$ & $-0.020(3)$  & $11.137$ & $-0.019(2)$  & $11.138$ \\\\ \\cline{4-8}\n                       &                      &                            & $7\/2$ & $0.015(2)$   & $11.172$ & $0.014(2)$   & $11.171$ \\\\ \\hline\\hline\n\\end{tabular}\n\\caption{Hyperfine contributions to the double bottom baryons for the $(1\/2)_u'$ static energy for two sets of parameters $\\Lambda_f$, $\\Lambda_f'$ of the hyperfine potentials ($\\Delta^{(0)}_{(1\/2)^{-}}=0.075$~GeV$^2$, $\\Delta^{(1,0)}_{(1\/2)^{-}}=\\Delta^{(1,2)}_{(1\/2)^{-}}=0$~GeV$^4$). All masses in GeV units.}\n\\label{hft12upb}\n\\end{table}\n\nWe plot the spectra for double charm and bottom baryons in Figs.~\\ref{ccplot} and \\ref{bbplot}, respectively, for the parameters of the entry $r_0=0.5$~fm in Table~\\ref{fits2p}.\n\n\\begin{figure}[ht!]\n   \\centerline{\\includegraphics[width=.6\\textwidth]{ccqspv2.pdf}}\n\t\\caption{Spectrum of double charm baryons in terms of $j^{\\eta_P}$ states. Each line represents a state. The spectrum corresponds to the results of Tables~\\ref{hft12gc} and \\ref{hft12upc} for states associated to the $(1\/2)_g$ and $(1\/2)_u'$ static energies and the results for Ref.~\\cite{Soto:2020pfa} for the mixed $(3\/2)_u\\backslash(1\/2)_u$ static energies, which do not include hyperfine contributions. The color indicates the static energies that generate each state.}\n\t\\label{ccplot}\n \\end{figure}\n\n\n\\begin{figure}[ht!]\n   \\centerline{\\includegraphics[width=.6\\textwidth]{bbqspv2.pdf}}\n\t\\caption{Spectrum of double bottom baryons in terms of $j^{\\eta_P}$ states. Each line represents a state. The spectrum corresponds to the results of Tables~\\ref{hft12gb} and \\ref{hft12upb} for states associated to the $(1\/2)_g$ and $(1\/2)_u'$ static energies and the results for Ref.~\\cite{Soto:2020pfa} for the mixed $(3\/2)_u\\backslash(1\/2)_u$ static energies, which do not include hyperfine contributions. The color indicates the static energies that generate each state.}\n\t\\label{bbplot}\n \\end{figure}\n\nLet us discuss the uncertainties of our results. The leading order masses, $M^{(0)}$, have uncertainties associated to the values of the heavy-quark masses and $\\overline{\\Lambda}_{(1\/2)^+}$, in Eqs.~\\eqref{mc}-\\eqref{lbar12}, as well as the uncertainty in the parametrization of the static potentials which was estimated as $10$~MeV in Ref.~\\cite{Soto:2020pfa}.  Adding these uncertainties in quadrature we obtain $\\delta M_{ccq}^{(0)}=39$~MeV and $\\delta M_{bbq}^{(0)}=48$~MeV. Furthermore, there is in principle an uncertainty related to the use of an unphysical light-quark mass in the lattice determinations of the static potentials of Refs.~\\cite{Najjar:2009da,Najjarthesis} that we used to obtain $M^{(0)}$ in Ref.~\\cite{Soto:2020pfa}. We expect the contribution due to the unphysical light-quark mass to be almost independent of $r$. This is supported by the calculations of the charmonium spectrum (with respect to the $\\eta_c$ mass) at $m_\\pi\\sim 400$ MeV \\cite{HadronSpectrum:2012gic} and $m_\\pi\\sim 240$ MeV \\cite{Cheung:2016bym}, in which almost no difference is observed for the masses of the states below threshold.\\footnote{An increase of the charmonium masses when the light-quark mass  decreases is observed for states about $1$~GeV higher than the $\\eta_c$ mass or beyond. If this is interpreted as due to an increase of the string tension with decreasing light-quark masses, then it is consistent with our findings in Appendix  \\ref{app:ce}, provided the mass of our fermion on the string is an increasing function of the light-quark mass.} Hence, it will just produce an overall shift to the static energies computed on the lattice. However, in the computation of Ref.~\\cite{Soto:2020pfa} the static energies were rescaled in order for the ground state static energy to be given in the short distance by the expression in Eq.~\\eqref{sdstp}. Therefore any additive constant contribution to the static energies produces no change in our results.\n\nThe hyperfine contribution, $M^{(1)}$, has uncertainties associated to the statistical errors of the values of the parameters and interpolation of the potentials. The former ones are displayed in parentheses in Tables~\\ref{hft12gc}-\\ref{hft12upb} and are about a few MeV for most cases, although in some instances larger values up to $9$~MeV can also be found. To assess the uncertainty associated to the choice of interpolation of the potentials in Eqs.~\\eqref{s1int}-\\eqref{lint} we take the difference of the hyperfine contributions computed with the parameter sets for $r_0=0.5$~fm of Table~\\ref{fits2p} and $r_0=0.3$~fm of Table~\\ref{fits4p}. This uncertainty of the hyperfine contribution amounts to $1$~MeV-$6$~MeV for double charm baryons and $1$~MeV-$4$~MeV for double bottom baryons except for a few cases in Tables~\\ref{hft12gc} and \\ref{hft12upc} for double charm states where the difference is larger. Finally, one should consider the size of higher-order contributions to the doubly heavy baryon masses. The most important is the contribution form heavy-quark-spin and angular-momentum independent $1\/m_Q$ suppressed potential of ${\\cal O}(\\Lambda^2_{\\rm QCD}\/m_Q)$, which we estimate as $\\sim 64$~MeV and $\\sim 19$~MeV for double charm and double bottom baryons, respectively. However, in the case of the hyperfine splittings the previous contribution cancels out and the higher order corrections correspond to the $1\/m^2_Q$ suppressed potentials of ${\\cal O}(\\Lambda^3_{\\rm QCD}\/m^2_Q)$, which we take as $\\sim 14$~MeV and $\\sim 1$~MeV for double charm and double bottom baryons, respectively.\n\nAs an example, in the following we show the value of the masses for the double charm ground state doublet, often refereed as $\\Xi_{cc}[(1\/2)^+]$ and $\\Xi^{*}_{cc}[(3\/2)^+]$, adding the different uncertainties in quadrature:\n\\begin{align}\n&m_{\\Xi_{cc}}=3.653(75)~{\\rm GeV}\\,,\\\\\n&m_{\\Xi^{*}_{cc}}=3.741(75)~{\\rm GeV}\\,,\n\\end{align}\nand for the double bottom ground state doublet:\n\\begin{align}\n&m_{\\Xi_{bb}}=10.120(52)~{\\rm GeV}\\,,\\\\\n&m_{\\Xi^{*}_{bb}}=10.150(52)~{\\rm GeV}\\,.\n\\end{align}\nIn the hyperfine splittings most of the uncertainties cancel out and hence our results have higher precision\n\\begin{align}\nm_{\\Xi^{*}_{cc}}-m_{\\Xi_{cc}}=88(14)~{\\rm MeV}\\,,\\label{hfsccn}\\\\\nm_{\\Xi^{*}_{bb}}-m_{\\Xi_{bb}}=30(5)~{\\rm MeV}\\,.\n\\end{align}\nThe figures above are compatible with all lattice determinations we are aware of, see Table~VI of Ref.~\\cite{Soto:2020pfa} and Table~\\ref{hfsbb} for doubly charmed and doubly bottom baryons respectively.\n\n\\begin{table}[ht!]\n\\begin{tabular}{||cc||}\\hline\\hline\nRef. & $\\delta_{hf}~[{\\rm MeV}]$     \\\\ \\hline\nOur value              & $30(5)$     \\\\\n\\cite{Brown:2014ena}   & $34.6(7.8)$ \\\\\n\\cite{Lewis:2008fu}    & $26(8)$     \\\\\n\\cite{Mohanta:2019mxo} & $32(5)$     \\\\ \\hline\\hline\n\\end{tabular}\n\\caption{Lattice results for the hyperfine splitting $\\delta_{hf}=M_{\\Xi^*_{bb}}-M_{\\Xi_{bb}}$.}\n\\label{hfsbb}\n\\end{table}\n\nLet us finally note that the hyperfine splittings of the $(1\/2)_u'$ states are entirely predicted from the long-distance parameters $\\Lambda_f$ and $\\Lambda_f'$, obtained from fits to the hyperfine splittings of the $(1\/2)_g$ states, and the only short-distance parameter, $\\Delta^{(0)}_{(1\/2)^-}$, obtained from the $D$-meson spectrum. It is then interesting to compare them with the lattice results of Ref.~\\cite{Padmanath:2015jea}. For the $(1\/2)_u'$ ground state doublet $(1\/2^-,3\/2^-)$, we obtain from Table~\\ref{hft12upc} $49(21)$~MeV for the hyperfine splitting, which agrees well with the $41(21)$~MeV of \\cite{Padmanath:2015jea}. This is a nontrivial test of the EST we use, since the $(1\/2)^+$ potentials differ from the $(1\/2)^-$ at long distances in a very particular way [see \\eqref{ldps1}-\\eqref{ldpl}]. For the  $(1\/2^+,3\/2^+)$ first angular excitation, we obtain $49(15)$~MeV, whereas there are two possible values from \\cite{Padmanath:2015jea} depending on how the state identifications are made, $25(58)$~MeV or $85(35)$~MeV, both of them compatible with our number within errors. State identification is plagued with ambiguities for higher excitations, which prevent us from making further comparisons. \n\n\\section{Comparison with models}\\label{sec:models}\n\nThere is a substantial amount of literature regarding doubly heavy baryons in different approaches; various quark models~\\cite{DeRujula:1975qlm,Ebert:1996ec,Gerasyuta:1999pc,Itoh:2000um,Gershtein:2000nx,Ebert:2002ig,Albertus:2006ya,Roberts:2007ni,Martynenko:2007je,Yoshida:2015tia,Kiselev:2017eic,Shah:2017liu,Weng:2018mmf,Lu:2017meb}, Bethe-Salpeter equations~\\cite{Migura:2006ep,Weng:2010rb,Li:2019ekr}, Born-Oppenheimer approximation with model potential~\\cite{Fleck:1989mb,Maiani:2019lpu}, semiempirical mass formulas~\\cite{Roncaglia:1995az,Karliner:2014gca,Lichtenberg:1995kg}, QCD sum rules~\\cite{Zhang:2008rt,Wang:2018lhz}, Faddeev equations~\\cite{Silvestre-Brac:1996tmn}, and bag models~\\cite{He:2004px}. In this section we compare our results with a selected set of model computations and other approaches (see \\cite{Shah:2017liu,Chen:2016spr,Mohajery:2018qhz} for further comparisons). In Table~\\ref{tab:mod:ccq} we have collected the masses of the ground state doublet in the double charm  baryon sector from different approaches. The values of the $\\Xi_{cc}$ mass are in good agreement for about $3\/4$ of the references, including our own value. Considering the uncertainties only a few works show very significant differences. The values for $\\Xi^{*}_{cc}$ show more dispersion with only half of the references being compatible with our own value. On the other hand, the splitting between the two masses is compatible with our value for only $1\/4$ of the references. This is in contrast with lattice QCD calculations, which are compatible with our current result for the hyperfine splitting (\\ref{hfsccn})  (see Table VI of ref. \\cite{Soto:2020pfa}).\n\n\\begin{table}[ht!]\n\\begin{tabular}{||c|c|c||} \\hline\\hline\nRef.                          & $\\Xi_{cc}[(1\/2)^+]$ & $\\Xi^{*}_{cc}[(3\/2)^+]$ \\\\ \\hline \nOur results                   & $3.653(75)$     & $3.741(75)$     \\\\\n\\cite{DeRujula:1975qlm}       & $3.550-3.760$   & $3.620-3.830$   \\\\\n\\cite{Fleck:1989mb}           & $3.613$         & $3.741$         \\\\\n\\cite{Roncaglia:1995az}       & $3.66(7)$       & $3.74(7)$       \\\\\n\\cite{Lichtenberg:1995kg}     & $3.676$         & $3.746$         \\\\\n\\cite{Ebert:1996ec}           & $3.660$         & $3.810$         \\\\\n\\cite{Silvestre-Brac:1996tmn} & $3.608$         & $3.701$         \\\\\n\\cite{Gerasyuta:1999pc}       & $3.527$         & $3.597$         \\\\\n\\cite{Itoh:2000um}            & $3.649(10)$     & $3.734(10)$     \\\\\n\\cite{Ebert:2002ig}           & $3.620$         & $3.727$         \\\\\n\\cite{He:2004px}              & $3.550$         & $3.590$         \\\\\n\\cite{Migura:2006ep}          & $3.642$         & $3.723$         \\\\\n\\cite{Albertus:2006ya}        & $3.612^{+(17)}$ & $3.706^{+(23)}$ \\\\\n\\cite{Roberts:2007ni}         & $3.676$         & $4.029$         \\\\\n\\cite{Martynenko:2007je}      & $3.510$         & $3.548$         \\\\\n\\cite{Zhang:2008rt}           & $4.26(19)$      & $3.9(1)$        \\\\\n\\cite{Karliner:2014gca}       & $3.627(12)$     & $3.690(12)$     \\\\\n\\cite{Yoshida:2015tia}        & $3.685$         & $3.754$         \\\\\n\\cite{Kiselev:2017eic}        & $3.615(55)$     & $3.747(55)$     \\\\\n\\cite{Shah:2017liu}           & $3.511$         & $3.687$         \\\\\n\\cite{Lu:2017meb}\t\t\t\t\t\t\t& $3.606$         & $3.675$          \\\\\n\\cite{Weng:2018mmf}\t\t\t\t\t\t& $3.633$         & $3.696$           \\\\\n\\cite{Wang:2018lhz}           & $3.630^{+(80)}_{-(70)}$ & $3.750(70)$ \\\\\n\\cite{Maiani:2019lpu}         & $3.621^{+(17)}_{-(7)}$ & -        \\\\ \n\\cite{Li:2019ekr}             & $3.601$         & $3.703$          \\\\ \\hline\nExp. \\cite{LHCb:2019epo}      & $3.6216(4)$   & - \\\\\n\\hline\\hline \n\\end{tabular}\n\\caption{Masses of double charm baryons from model computations in GeV units.}\n\\label{tab:mod:ccq}\n\\end{table}\n\nThe masses of the ground state doublet in the double bottom baryon sector are shown in Table~\\ref{tab:mod:bbq}. In this case the  differences are a lot more significant. For both the $\\Xi_{bb}$ and $\\Xi^{*}_{bb}$ only Refs.~\\cite{He:2004px,Martynenko:2007je,Karliner:2014gca,Gershtein:2000nx} are compatible with our results and in general there is more dispersion among the values of the different model approaches. Although the values of the hyperfine splittings present less variation in absolute values, in relative terms the variation is also larger than in the double charm baryon sector. Moreover, very few values are compatible with ours. This is due to our small uncertainty for the splitting produced by the cancellation of uncertainties associated to various parameters remaining only the uncertainty on higher order contributions, which are small for double bottom baryons. We note that no reference has compatible results with ours for both for the $\\Xi_{bb}$ and $\\Xi^{*}_{bb}$ masses and the hyperfine splitting. This is in contrast with the good agreement we get with the available lattice results (see Table \\ref{hfsbb}).\n\n\\begin{table}[ht!]\n\\begin{tabular}{||c|c|c||} \\hline\\hline\nRef.                          & $\\Xi_{bb}[(1\/2)^+]$      & $\\Xi^{*}_{bb}[(3\/2)^+]$ \\\\ \\hline\nOur results                   & $10.120(52)$             & $10.150(52)$            \\\\\n\\cite{Roncaglia:1995az}       & $10.34(10)$              & $10.37(10)$             \\\\\n\\cite{Ebert:1996ec}           & $10.23$                  & $10.28$                 \\\\\n\\cite{Silvestre-Brac:1996tmn} & $10.198$                 & $10.236$                \\\\\n\\cite{Gershtein:2000nx}       & $10.093$                 & $10.133$                \\\\\n\\cite{Ebert:2002ig}           & $10.202$                 & $10.237$                \\\\\n\\cite{He:2004px}              & $10.10$                  & $10.11$                 \\\\\n\\cite{Albertus:2006ya}        & $10.197^{+(10)}_{-(17)}$ & $10.236^{+(9)}_{-(17)}$ \\\\\n\\cite{Roberts:2007ni}         & $10.340$                 & $10.367$                \\\\\n\\cite{Martynenko:2007je}      & $10.130$                 & $10.144$                \\\\\n\\cite{Zhang:2008rt}           & $9.78(7)$                & $10.28(5)$              \\\\\n\\cite{Karliner:2014gca}       & $10.162(12)$             & $10.184(12)$            \\\\\n\\cite{Yoshida:2015tia}        & $10.314$                 & $10.339$                \\\\\n\\cite{Shah:2017liu}           & $10.312$                 & $10.335$                \\\\ \n\\cite{Li:2019ekr}             & $10.182$                 & $10.214$                \\\\\n\\cite{Weng:2018mmf}\t\t\t\t\t\t& $10.169$                 & $10.189$                \\\\\n\\cite{Lu:2017meb}\t\t\t\t\t\t\t& $10.138$                 & $10.169$                \\\\\n\\cite{Wang:2018lhz}           & $10.220(70)$             & $10.270(70)$            \\\\\n\\hline\\hline  \n\\end{tabular}\n\\caption{Masses of double bottom baryons from model computations in GeV units.}\n\\label{tab:mod:bbq}\n\\end{table}\n\nThe spectrum of doubly heavy baryons beyond the ground state doublet has also been studied in Refs.~\\cite{Gershtein:2000nx,Ebert:2002ig,Yoshida:2015tia,Shah:2017liu,Lu:2017meb,Li:2019ekr}. In Fig.~\\ref{comp_all}  we compare our spectra with the ones in Ref.~\\cite{Ebert:2002ig,Yoshida:2015tia,Lu:2017meb} obtained with a quark model with a relativistic light quark, a nonrelativistic quark model, and a relativistic quark model with a diquark core respectively. The spectra of Ref.~\\cite{Gershtein:2000nx} is derived from a similar quark model as in Ref.~\\cite{Ebert:2002ig}, but the values are shifted down by about a $100$~MeV. Ref.~\\cite{Li:2019ekr} uses the Bethe-Salpeter equation in a diquark picture and presents a limited number of states in the spin-symmetry limit. The results of Ref.~\\cite{Shah:2017liu} do not include the Pauli principle for the heavy-quark wave functions and we do not consider it beyond the ground state. From Fig.~\\ref{comp_all} (a) we can see that for double charm baryons the pattern of states beyond the ground state doublet does not agree with ours in none of the cases or among the quark model approaches themselves. For all displayed model spectra the excited states lie (much) lower than ours. This is in contrast to the overall agreement found with lattice calculations in~\\cite{Soto:2020xpm}. For double bottom  baryons [see Fig.~\\ref{comp_all} (b)] the discrepancies reach the ground state doublet, as the results of Ref.~\\cite{Ebert:2002ig}, and to a lesser extend the ones of Ref.~\\cite{Yoshida:2015tia}, lie higher than ours. However, there is agreement for the first excited (odd-parity) doublet, except for Ref.~\\cite{Ebert:2002ig}. For higher states the discrepancies persist, except for the odd-parity states of Ref.~\\cite{Lu:2017meb}, which are compatible with ours. \n\n\\begin{figure}[ht!]\n\\begin{tabular}{cc}\n\\includegraphics[width=.45\\textwidth]{ccqsall.pdf} & \\includegraphics[width=.45\\textwidth]{bbqsall.pdf}\\\\ \n(a) & (b) \\\\\n\\end{tabular} \n\\caption{Comparison of our results with those of Refs.~\\cite{Ebert:2002ig,Yoshida:2015tia,Lu:2017meb} (green, red, orange) for double charm and bottom baryons in (a) and (b), respectively. Our results (blue) correspond to the entries $r_0=0.5$~fm in Tables~\\ref{hft12gc} - \\ref{hft12upb}.}\n\\label{comp_all}\n\\end{figure}\n\n\\section{Conclusions}\\label{sec:con}\n\nAn EFT describing doubly heavy hadrons was put forward in Ref.~\\cite{Soto:2020xpm}. It is built upon the nonrelativistic expansion of the heavy quarks and the adiabatic expansion between the dynamics of the heavy quarks and the light degrees of freedom corresponding to the gluons and light quarks. The EFT was constructed in the single hadron sector up to the heavy-quark spin and angular momentum terms suppressed by $1\/m_Q$. Expressions of the potentials as operator insertions in the Wilson loop were obtained by matching the EFT to NRQCD. The computation of the Wilson loop with operator insertions cannot be done using perturbative techniques and should be carried out (ideally) in lattice QCD or other nonperturbative approaches (see for instance \\cite{Andreev:2020xor} for an AdS\/CFT inspired proposal).\n\nIn Ref.~\\cite{Soto:2020pfa} this EFT framework was applied to doubly heavy baryons. Using the lattice data of Refs.~\\cite{Najjar:2009da,Najjarthesis} for the static energies the leading-order spectrum of doubly charm and bottom baryons was computed for the four lowest lying static states. However, since there are no available lattice determinations of the potentials of the heavy-quark spin and angular-momentum operators, the computation of the hyperfine contributions to the doubly heavy baryon masses was not possible.\n\nIn this paper we presented a parametrization of the $1\/m_Q$ suppressed heavy-quark spin and angular-momentum operators with a minimal amount of modeling, the general idea of which can be extended to other potentials for doubly heavy hadrons, such as double charm tetraquark, $T^+_{cc}$, recently discovered by the LHCb Collaboration \\cite{LHCb:2021vvq}. This parametrization of the potentials is based in their description in short- and long-distance regimes. In the short-distance regime, defined as $r\\ll 1\/\\Lambda_{\\rm QCD}$, the Wilson-loop expressions of the potentials can be expanded in the multipole expansion. This can be done using weakly-coupled pNRQCD, which is the EFT that incorporates the multipole expansion systematically, for two heavy quarks~\\cite{Brambilla:2005yk}. This produces short-distance expressions of the potentials as an expansion in powers of $r^2$ with coefficients that encode the nonperturbative dynamics of the light degrees of freedom,\\footnote{In general the potentials can have a nonanalytical term in $r$ originating from the perturbative integration of the heavy-quark momentum, i.e when the potential can be generated without interacting with the light degrees of freedom. However, this is not the case for the potentials of the $1\/m_Q$ suppressed heavy-quark spin and angular momentum operators.} which we show in Sec.~\\ref{sec:sdp} and Appendix~\\ref{app:sdec}. At leading order in the multipole expansion only one coefficient is necessary and it can be determined in a model-independent way using the heavy quark-diquark duality from the heavy-mesons masses.\n\nThe long-distance regime is characterized by $r\\gg 1\/\\Lambda_{\\rm QCD}$. In the case of a heavy-quark-antiquark pair it is known from lattice QCD  that in this regime a gluonic flux tube connecting the two heavy quarks emerges. It is well-known that an Effective String Theory (EST)~\\cite{Nambu:1978bd,Luscher:1980fr,Luscher:2002qv} reproduces accurately the lattice determinations~\\cite{PerezNadal:2008vm,Hwang:2018rju}. In Sec.~\\ref{sec:est} we propose an extension of the EST to include the presence of a fermion constrained to move on the string. We obtain a mapping of the NRQCD operators inserted in the Wilson loop to operators in the EST based on imposing the same transformation properties under $D_{\\infty h}$ and flavor. Using this mapping we can translate the Wilson-loop expressions for the potentials to EST correlators and evaluate them. This procedure yields long-distance expressions of the potentials depending on two unknown coefficients of the EST. Additionally, we compute the vacuum  energy in the EST with fermions in Appendix~\\ref{app:ce} and show that (i) the string tension runs with the square of the mass of the fermion and (ii) the sign of the L\\\"uscher term changes. These features can in principle be checked by lattice calculations of the ground state energy of two static quarks separated at a large distance with an additional light quark.\\footnote{Beyond the string breaking scale, it would also require the calculation of excitations with the ground state quantum numbers.}\n\nThe final parametrization of the potentials is obtained by interpolating between the short- and long-distance descriptions. We choose the most simple interpolation that ensures that the correct short- and long-distance behavior are recovered in the corresponding limits. Nevertheless, an extra parameter is introduced in the definition of the interpolation. The hyperfine contributions to doubly heavy baryons can be computed using these parametrizations of the heavy-quark spin and angular-momentum dependent potentials. The values of the remaining unknown parameters are determined by fitting the hyperfine splittings obtained in lattice QCD in Refs.~\\cite{Briceno:2012wt,Namekawa:2013vu,Brown:2014ena,Alexandrou:2014sha,Bali:2015lka,Padmanath:2015jea,Alexandrou:2017xwd,Lewis:2008fu,Brown:2014ena,Mohanta:2019mxo} for several $S$-, $P$- and , $D$-wave multiplets. This guarantees that all our inputs are from QCD, and the modeling is reduced to the choice of interpolation, provided that the EST we use is the correct effective theory at long distances. Using the parameters thus determined, we make predictions for the spectrum of double charm and bottom baryons including the hyperfine contributions. Our results are summarized in Tables~\\ref{hft12gc}-\\ref{hft12upb} and in Figs.~\\ref{ccplot} and \\ref{bbplot}. \n\nFinally in Sec.~\\ref{sec:models} we compared our results with previous model approaches and sum rules determinations. We observe a huge dispersion of results. In the absence of lattice calculations for many states, specially for double bottom baryons, our EFT approach offers a framework in which modeling is minimal and errors can be reliably quantified, unlike in most models. Since lattice determinations of the potentials for doubly heavy hadrons are difficult, in particular when unquenched simulations are required, the procedure outlined in the paper and in Ref.~\\cite{Oncala:2017hop}, to obtain reliable parametrizations of the potentials can be of significant utility in future studies of doubly heavy hadrons. In turn, this motivates further development of the EST to cases with multiple light quarks.\n\n\\section*{Acknowledgments}\n\nJ.S. acknowledges financial support from Grant No.~2017-SGR-929 from the Generalitat de Catalunya and from projects No.~PID2019-\n105614GB-C21, No.~PID2019-110165GB-I00 and No.~CEX2019-000918-M from Ministerio de Ciencia, Innovaci\\'on y Universidades.  J.T.C acknowledges the financial support from the European Union's Horizon 2020 research and innovation program under the Marie Sk\\l{}odowska--Curie Grant Agreement No. 665919. He has also been supported in part by the Spanish Grants No.~FPA2017-86989-P and No.~SEV-2016-0588 from the Ministerio de Ciencia, Innovaci\\'on y Universidades, and Grants No.~2017-SGR-1069 from the Generalitat de Catalunya. This research was supported by the Munich Institute for Astro and Particle Physics (MIAPP) which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy No.~EXC-2094\u2013390783311.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sect:intro}\nNGC\\,7213 (z=0.005839) is a face on Sa galaxy hosting an active galactic nucleus (AGN). It has been classified as a type I Seyfert by \\citet{phillips79}, based on its $H\\alpha$ linewidth (full width at zero intensity of $13000$ km s$^{-1}$), but also as a low-ionization nuclear emission-line region (LINER) by \\citet{filippenko84},  based on the study of a  variety of optical emission lines which were observed to have a full width at half maximum of  200 to 2000 km s$^{-1}$.\\par\nNGC\\,7213 hosts a black-hole with a mass of about $10^8\\;{\\rm M}_{\\sun}$ \\citep{woo02}. Its  bolometric luminosity of $L_{\\rm bol}=9\\times10^{42}$ erg s$^{-1}$\\citep{starling05}  suggests a rather low accretion rate of 0.07 per cent of the Eddington luminosity (L$_{\\rm Edd}$). As noted by \\citet{lobban10}, this value is intermediate between those usually found in type I Seyfert galaxies and LINER's and it is significantly less than the predicted 2 per cent L$_{\\rm Edd}$ `critical' rate at which the `soft state' transition appears in black hole X-ray binaries (BHXRBs) \\citep{maccarone03}. Finally, based on its radio properties, NGC\\,7213 belongs to a rare class of extragalactic sources lying between the radio loud and radio quiet AGN (it is one of the 20 sources of the \\citet{roy97} sample). In this sense, it can be considered as an extragalactic analogue of the Galactic `hard state' sources.\\par\nThe X-ray spectral behaviour NGC\\,7213 is also peculiar. \\citet{starling05} using the reflection grating spectrometer, on board \\textit{XMM-Newton}, found several emission features with no absorption lines; contrary to what is usually observed in type I Seyfert galaxies. The absence of a Compton reflection component from either neutral or ionized material together with the lack of a relativistic Fe K$\\alpha$~ line suggests that the inner, optically thick accretion disc in the source may be absent \\citep{lobban10}, and replaced by an advection-dominated accretion flow (ADAF) \\citep{starling05}. This possibility further supports the `hard state' interpretation of the source.\\par\nRecently, \\citet{bell11} used radio and X-ray observations of NGC\\,7213, from a long-term observing campaign consisting of several years, to search for correlated X-ray and radio variations that might originate in the so-called `fundamental plane of black hole accretion' \\citep{merloni03,koerding06}. \\citet{bell11} showed that the average radio and X-ray luminosities fitted well with the global fundamental plane, consistent with a `hard state' identification of this AGN. However, the X-ray versus radio correlation within the monitoring period was weak, with significant intrinsic scatter away from the plane.\\par\nNumerous studies in the past have shown that the X-ray photon index, $\\Gamma$, correlates with the accretion rate, defined as $\\xi=L_{\\rm X}\/L_{\\rm Edd}$ (where $L_{\\rm X}$ is usually the 2--10 keV X-ray luminosity) in both AGN and BHXRBs. For example, \\citet{shemmer06} showed that $\\Gamma$ and $\\xi$ are positively correlated, using data for a sample of thirty quasars, while \\citet{sobolewska09} found that the same positive correlation holds when one studies the spectral variations of individual, luminous Seyfert galaxies. However, the positive $\\Gamma-\\xi$ correlation may not hold in less luminous AGN. In fact \\citet{gu09} found an anti-correlation between $\\Gamma$ and $\\xi$ in a sample of 55 low-luminosity AGN (LLAGN), and \\citet{younes11} reached a similar conclusion for a sample of 13 optically selected LINER's.\\par\n\\citet{wu08} performed a detailed spectral study for six BHXRBs and found that $\\Gamma$ anti-correlates with $\\xi$ below a `critical' value of $\\log(\\xi_{\\rm crit})=-2.1\\pm0.2$. At higher accretion rates, $\\Gamma$ and $\\xi$ are positively correlated. Similar conclusions were also reached recently by \\citet{sobolewska11}. The $\\Gamma$ -- $\\xi$ relation in AGN may also change from negative to positive at a similar `critical' value, as it was suggested by \\citet{wu08}, and \\citet{constantin09}. This analogous behaviour between AGN and BHXRBs reinforces the interpretation that LLAGN are analogues of the `hard state' BHXRBs while luminous Seyferts and quasars are the equivalent of the `soft state' BHXRBs. On a first look, during the `hard state', BHXRBs appear to have a constant hardness-ratio, in the commonly used hardness-intensity diagrams \\citep[e.g.][]{belloni05}. Nevertheless, detailed analysis of the lowest luminosity regime of this state, known also as `low-hard' state, exhibits clearly an increase of the hardness-ratio for increased fluxes, corresponding to an anti-correlation between $\\Gamma$ and $\\xi$ \\citep{heil12}.\\par\nThe negative and the positive correlations between $\\Gamma$ and $\\xi$, occurring below and above $\\xi_{\\rm crit}$, respectively, may be indicative for some `switch' in the emission mechanism as the source's accretion rate increases above $\\xi_{\\rm crit}$. In fact, it has been suggested \\citep{wu08,constantin09,younes11} that this transition could indicate the passage from an ADAF \\citep{narayan94,esin97} to standard disc (but see \\citet{sobolewska11} for alternative explanations as well).\\par\nThe positive correlation relation between $\\Gamma$ versus $\\xi$ in AGN has been firmly established statistically by either considering samples of numerous sources \\citep[e.g.][]{shemmer06} or large data sets for a few individual sources \\citep[e.g.][]{mchardy99,larmer03,sobolewska09}. However, up to now the $\\Gamma$ -- $\\xi$ anti-correlation has been established using only short X-ray observations of numerous different sources \\citep{gu09,constantin09,younes11}, while it has never been observed in a single source.\\par\nIn this work, we use long-term, monitoring \\textit{RXTE} data, and present, for the first time, conclusive evidence for such an anti-correlation for an individual LLAGN, namely NGC\\,7213. We also use archival and proprietary data to construct its average spectral energy distribution (SED), and compare it with the average SED of luminous AGN and LINER's. In Section \\ref{sect:obs_reduc} we refer to the observations and data reduction procedures and in Section \\ref{ssect:rxte_lc} we present our results consisting of the $\\Gamma$ versus flux relation and the spectral energy distribution of the source (SED) of the source. Finally, a discussion together with a summary can be found in Section \\ref{sect:discussion}. The cosmological parameters used throughout this paper are: $\\rm{H}_0=70$ km s$^{-1}$ Mpc$^{-1}$, $\\Omega_{\\Lambda}$=0.73 and $\\Omega_{\\rm m}$=0.27, yielding a luminosity distance to NGC\\,7213 of 22.12 Mpc (for a corrected redshift, $z_{\\rm corr.3K}=0.005145$ to the reference frame defined by the 3 K cosmic microwave background radiation). This value of the luminosity distance appears to be fully consistent\nwith the redshift-independent Tully-Fisher distance \\citep{tully88}.\n \n\\section{OBSERVATIONS AND DATA-REDUCTION}\n\\label{sect:obs_reduc}\n\\subsection{X-ray data}\nNGC\\,7213 has been observed regularly by the \\textit{Rossi X-ray Timing Explorer} (\\textit{RXTE}) (proposal numbers: 92119, 93139, 94139 and 94342) from 2006 March 3, 02:57:41 (UTC) to 2009 December 30, 00:18:37 (UTC) (on-time: 800506 s). This is the complete set of X-ray observations of NGC\\,7213, obtained by RXTE, extending the X-ray data of \\citet{bell11} by an additional of 100 days. In order to form the most homogeneous long-term data set, we use data obtained only by the proportional counter array 2 (PCU\\,2), one of the five Xenon proportional counters forming the proportional counter array \\citep{jahoda96}. PCU\\,0 and 1 have lost their propane layer, resulting an increased background rate as well as a different detector gain, and PCU\\,3 and 4 are usually switched off due to discharge problems.\\par\nThe {\\tt Standard-2} (Std2: data with accumulation rate of 16 sec) PCU\\,2 data (822 files in total) are extracted and processed using {\\tt ftools} \\citep{blackburn95} included in {\\tt HEAsoft} (ver.~6.11.1), following the standard reduction procedures, provided by the \\textit{RXTE} Guest Observer Facility (GOF). For each observation we create a filter file, assembling the scientifically important parameters, using the script {\\sc xtefilt} and then by using the {\\tt ftool} {\\sc fmerge} we merge them all in a single `master' filter file. Then, based on this file, we extract the useful observing time periods, known as good time intervals (GTIs), using the {\\tt ftool} {\\sc maketime} having the following observational constraints: elevation angle greater than 10$\\degr$, a pointing offset of less than 0.02$\\degr$, electron contamination less than 0.1, time since south Atlantic anomaly between zero and thirty minutes and time since PCU breakdowns less than -150 s or greater than 600 s. Finally, for each Std2 observation a synthetic background model file is created with the {\\tt ftool} {\\sc pcabackest} using the `faint background model'.\\par\nFor the production of the background-subtracted light curves, in the 2--4, 5--10 and 2--10 keV energy bands, we use the {\\tt ftool} {\\sc saextrct} to produce the `source-plus-background' and `background' light curves for each energy band, respectively. We select events obtained during the GTIs registered only from the top xenon layer (X1L, X1R) of PCU\\,2, optimizing in this way the signal to noise ratio. Then the corrected background-subtracted light curves are produced by using the {\\tt ftool} {\\sc lcmath} in units of count-rate. In order to produce a total long-term X-ray light curve in the 2--10 keV energy range, in units of flux i.e.\\ erg s$^{-1}$ cm$^{-2}$, we extract for each Std2 observation, and the corresponding synthetic background file, a spectrum together with the PCU\\,2's response using again the {\\tt ftool} {\\sc saextrct} and the script {\\sc pcarsp}, respectively. Again, we select events obtained during the GTIs which are registered only from the top xenon layer of PCU\\,2. The resulting spectra are then grouped so that each spectral bin contains at least 30 counts, using the {\\tt ftool} task {\\sc grppha}.\\par \nUsing the X-ray fitting package {\\tt XSPEC} (ver.~12.7.0) \\citep{arnaud96} we fit to each X-ray spectrum a power law model assuming photoelectric absorption ({\\tt XSPEC} model {\\tt wabs}), of a fixed interstellar column-density of $N_{\\rm H}=1.1\\times 10^{20}$ cm$^{-2}$ \\citep[estimated using the {\\sc ftool} \\textit{nH}, after][]{kalberla05} . Finally, after fixing the fitting parameters (i.e.\\ photon index and normalisation) to their the best-fitting values, we obtain the flux values, in the 2--10 keV energy range using the convolution model {\\tt cflux}. The errors in the spectral indices and fluxes indicate their 68.3 per cent confidence range, corresponding to a $\\Delta \\chi^2$ of 1, unless otherwise stated.\n\n\\subsection{Spectral energy distribution data}\n\\label{ssect:sed_data}\nIn order to construct the SED of NGC\\,7213, we analyse long term monitoring radio and optical data. Below we describe the data analysis procedures for these observations. We also use the published near-infrared, nuclear fluxes of \\citet{hoenig10,asmus11} at 12.27 and 11.25 and 10.49 $\\mu$m (these fluxes are listed in Table \\ref{tab:fluxEntries} with an asterisk). Finally, we also considered the results from a $130$ ksec \\textit{XMM-Newton} observation obtained during 55146--55148 MJD (obs: ID-0605800301, Emmanoulopoulos et al.\\,in prep.), the 4.5 years average \\textit{Swift}-BAT spectrum \\citep[as reported in entry 1185 of table 2 in][]{cusumano10}, and the recent 0.1--100 GeV \\textit{Fermi-}LAT upper limit \\citep{lobban10}.\n\n\\begin{table}\n\\caption{The radio, near-infrared and optical mean flux values of NGC\\,7213.}\n\\label{tab:fluxEntries}\n\\centering\\begin{tabular}{@{}cccc}\n\\hline\n Energy band & Flux \\\\\n & mJy \\\\\n\\hline\n 1.344 GHz  & $121.3 \\pm 2.2$  \\\\\n 1.384 GHz & $112.4\\pm 9.3$    \\\\\n 2.386 GHz  & $114.9 \\pm 3.6$    \\\\\n 2.496 GHz  & $98.1 \\pm 4.1$    \\\\\n 4.8 GHz & $135.8 \\pm 8.3$       \\\\\n 8.64 GHz  & $150.6 \\pm 24.3$     \\\\\n 17 GHz  & $140.4 \\pm 3.9$     \\\\\n 19 GHz  & $128.5 \\pm 4.1$      \\\\\n 12.27 $\\mu$m\\textsuperscript{\\textasteriskcentered} & $235.8 \\pm 128.4$ \\\\ \n 11.25 $\\mu$m\\textsuperscript{\\textasteriskcentered} & $232.9 \\pm 144.5$ \\\\\n 10.49 $\\mu$m\\textsuperscript{\\textasteriskcentered} & $239.1 \\pm 22$ \\\\\n 5500 \\AA &  $0.49 \\pm 0.18$ \\\\\n 4400 \\AA & $0.73 \\pm 0.15$  \\\\\n\\hline\n\\end{tabular}\n\\medskip \\\\\n\\textsuperscript{\\textasteriskcentered} the near-IR measurements taken from \\citet{hoenig10,asmus11} as explained in the text.\n\\end{table}\n\n\\subsubsection{Radio data}\nWe searched the Australian Telescope Online Archive (ATOA) for \\textit{Australian Telescope Compact Array} (\\textit{ATCA}) observations of NGC\\,7213's location, selecting a range that covered as many frequencies as possible, opting for the longest duration observation(s) in each case (sometimes yielding multiple useful files, as in the case of 5--10 GHz). While attempts were made to process all designated files, the highest available frequencies, greater than 20 GHz, did not yield useful images. The final data set used in this work was comprised of 21 ATCA observations, from four projects; C782, C1803, C1532 and C1392. \\par\nThe primary calibrator for all these data sets was PKS\\,1939-6342 (PKS\\,B1934-638). C782 includes observations from 1999 March 03 and frequencies of 1344 MHz (0.3 h on source, average beam size (ABS): $64.7\\arcsec\\times7.0\\arcsec$), 1384 MHz (8.5 h on source, ABS: $21.6\\arcsec\\times7.2\\arcsec$) and 2496 MHz (8.8 h on source, ABS: $7.6\\arcsec\\times4.9\\arcsec$ ), with the antennae in configuration 6C (minimum baseline length of 153 m, maximum of 6 km). The secondary calibrator used was PJS\\,J2214-3835 (PKS\\,B2211-388A). C1803 included observations from 2008 April 21 and 23 at 1384 MHz and 2386 MHz, with 0.15 and 0.1 h on source respectively. The antenna configuration was 6A (minimum baseline length of 337 m, maximum of 5939 m) and the secondary calibrator used was PKS\\,J2235-4835 (PKS\\,B2232-488). C1532 contributed all our 4800 MHz (ABS: $17.3\\arcsec\\times1.8\\arcsec$) and 8640 MHz (ABS: $9.64\\arcsec\\times1\\arcsec$) data, with observations covering the dates of 2008 January 05, 24, April 12, November 04 and 20 with on source durations of 1.48, 1.7, 1.5, 2.29 and 1.49 h, respectively. The secondary calibrator used in all cases was PKS\\,J2218-5038 (PKS\\,B2215-508) and antennae were always in configuration 6A. Finally, C1392 provided our highest frequency ATCA data, at 17 GHz (ABS: $35.5\\arcsec\\times27.55\\arcsec$) and 19 GHz (ABS: $29.15\\arcsec\\times22.8\\arcsec$). Two observations are used, from 2009 September 29 and 30 with 0.02 and 0.03 h on source respectively. The secondary calibrator was PKS\\,J2248-3235 (PKS\\,B2245-328) and the antenna configuration was H75 (minimum baseline of 31m, maximum of 4408m).\\par \nOnce imaging was complete, point-source fitting is used to measure flux density at NGC\\,7213's position $\\rmn{RA}(1950)=22^{\\rmn{h}}~09^{\\rmn{m}}~16\\fs26$, $\\rmn{Dec.}~(1950)=-47\\degr~09\\arcmin~59\\farcs 95$. All data and image processing is carried out in the radio interferometry data reduction package {\\sc MIRIAD} \\citep{sault95}.\n\n\\subsubsection{Optical data}\nWe used the \\textit{ANDICAM} instrument mounted on the 1.3 m telescope \\textit{SMARTS} in Chile to observe NGC\\,7213 in optical B (4400 \\AA) and V (5500 \\AA) bands. Observations were performed between August 2006 and August 2011, every 4 days and during each observations two 60 s exposure frames in the B band and two 30 s exposures in the V band were obtained when the target was visible. After discarding the `bad seeing' nights, a total of 266 epochs are accumulated in each band. The field of view of \\textit{ANDICAM} is $6\\arcmin\\times 6\\arcmin$ covering both the entire host galaxy and a few bright stars, and the pixel size is $0.37\\arcsec$ per pixel. The typical seeing during the campaign was about $1.5\\arcsec$. The data reduction for both the B and V band images is the same.\\par \nSince NGC\\,7213 host galaxy is a face-on spiral with a bright nucleus compared to the AGN, galaxy subtraction has to be performed to determine the pure AGN flux. To model correctly the large scale structure we use the data analysis algorithm {\\tt GALFIT} (ver.~3.0) \\citep{peng10}, fitting a Nuker function. After the fitting, a bright ring $10\\arcsec$ from the nucleus is left together with some spiral structure inward of the ring and a resolved core. The core is subtracted using a Moffat function leaving a residual nuclear peak consistent with the stellar point-spread function (PSF), which we identified with the AGN. Aperture photometry is performed with the task {\\sc daophot}, of the {\\tt IRAF} (ver.~2.15) software system \\citep{tody93}, on the stars of the original image and on the modelled nuclear PSF to determine its relative brightness. In order to estimate the flux, the reference stars in turn are calibrated taking their average aperture magnitudes over all reported photometric nights when observations were carried out, corrected for airmass extinction. The average nuclear fluxes that are obtained in this way are $5\\times10^{-12}$ erg s$^{-1}$ cm$^{-2}$ at 4400 \\AA\\ and $3\\times10^{-12}$ erg s$^{-1}$ cm$^{-2}$ at 5500 \\AA, while the fractional root mean square variability amplitude (including observational noise and intrinsic variability) is around 1 erg s$^{-1}$ cm$^{-2}$ for both bands, respectively.\n\n\\section{RESULTS FOR NGC\\,7213}\n\\begin{figure}\n\\includegraphics[width=3.5in]{figures\/rxte_lc_2_10.eps}\\\\[-1.57em]\n\\hspace*{0.1em}\\includegraphics[width=3.555in]{figures\/rxte_photIndex_2_10.eps}\n\\caption{The RXTE results of NGC\\,7213. (Top panel) The long term light curve in the 2--10 keV energy range. The double-headed arrow indicates a period intense monitoring between 54982--55077 MJD. (Lower panel) The photon index of each pointing observation as a function of time.}\n\\label{fig:rxte_lc_pi}\n\\end{figure}\n\\subsection{The X-ray flux -- photon index relation in NGC\\,7213}\n\\label{ssect:rxte_lc}\nThe upper panel of Fig.~\\ref{fig:rxte_lc_pi} shows the 2--10 keV, X-ray light curve of NGC\\,7213. One can observe variations on time scales of days and weeks, superimposed on a flux decreasing trend, from the start to the end of the monitoring campaign. Each measurement on this plot corresponds to one \\textit{RXTE} pointing, having a mean duration of $1.06\\pm0.25$ ks (the error estimate corresponds to the standard deviation of the exposure times of individual observations). On average \\textit{RXTE} observed the source every $\\sim 2.3$ days, except from the period between 54982--55077 MJD  (indicated by the double-headed arrow in the upper panel of Fig.~\\ref{fig:rxte_lc_pi}), where observations were performed around every half a day. Observations with large uncertainties e.g.\\ around (53797+450) MJD, correspond to exposure times less than 0.9 ks.\\par \nThe lower panel of Fig.~\\ref{fig:rxte_lc_pi} shows the evolution of the $\\Gamma$ as a function of time. In order to investigate any possible relationship between flux and $\\Gamma$, we group both data sets in bins of 45 consecutive observations (each bin is then $\\sim100$ days long). For the intense monitoring period (1185--1280 MJD-53797) we apply the same sampling scheme by selecting only the observations that are separated around 2.3 days and then bin them in bins of 45 consecutive observations. In this way we ensure that the resulting X-ray flux and photon index range are represented by equal number of observations, avoiding the possibility that the larger number of observations in the intense monitoring period could drive the resulting flux\/photon index relation. \nWe estimate a weighted mean and a weighted error \\citep[e.g.][]{bevington92} for both flux and $\\Gamma$ in each bin. Fig.~\\ref{fig:rxte_hard_bright} shows the resulting average  X--ray flux and $\\Gamma$ values, plotted against each other, unveiling a clear `harder when brighter' behaviour.\\par\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{figures\/rxte_harder_brighter.eps}\\\\[-1.6em]\n\\caption{The average photon index versus the average X-ray flux of NGC\\,7213 in bins of 45 consecutive observations. The data suggest a `harder when brighter' relation. The solid line indicates the best-fit linear model to the data taking into account the uncertainties in both coordinates.}\n\\label{fig:rxte_hard_bright}\n\\end{figure}\n\nIn order to quantify this anti-correlation we first compute the Kendall's $\\tau$ rank correlation coefficient \\citep{press92a}, using the data plotted in Fig.~\\ref{fig:rxte_hard_bright}, yielding a $\\tau=-0.81$ with a null hypothesis probability of $2.6\\times 10^{-5}$. This results shows that the anti-correlation between spectral slope and flux is highly significant. Then, we fit to the data a linear model, $(y=\\alpha x+\\beta)$, considering the uncertainties in both coordinates \\citep{press92a}. The best-fitting model yields a slope of $\\alpha=-0.063\\pm0.018$ and an intercept of $\\beta=1.99\\pm0.04$ with a $\\chi^2$ merit function of 2.90 for 13 degrees of freedom (d.o.f.) having a null hypothesis probability of $1.7\\times10^{-3}$. Fig.~\\ref{fig:rxte_linRegrConf} shows the confidence contour (solid black line) used to estimate the one-standard deviation uncertainties on the best-fitting parameters corresponding to an ellipsoidal region with a $\\chi^2$ of 3.90 ($\\Delta\\chi^2=1$ from the minimum for 1 d.o.f.). In the same plot, we show also the contours for confidences of 95 and 99 per cent as well as the 68.3 per cent joint confidence contour for both the slope and the intercept with the latter corresponding to a $\\Delta\\chi^2=2.30$ from the minimum for 2 d.o.f. A simple linear regression model, taking into account only the photon index uncertainties, yields equivalent results: $\\alpha=-0.063\\pm0.008$ and $\\beta=1.99\\pm0.02$ ($\\chi^2$=2.90 for 13 d.o.f). The resulting $\\chi^2$, from both methods, are relatively small, indicating that the estimated errors on the average $\\Gamma$ within each bin maybe slightly overestimated. We therefore repeat the fit and this time we fit to the data a linear model following the `ordinary least-squares regression of Y on X' routine of \\citet{isobe90} which does not take into account the error on the data. The best-fit results from these routines are consistent, within the quoted errors, with the previous results yielding:  $\\alpha=-0.072\\pm 0.008$ and $\\beta=2.00\\pm 0.02$.\\par\nWe have performed several sanity checks in order to test the sensitivity of our results to the binning scheme. We have considered alternative binning of 20,60 and 100 consecutive observations as well as considering all the data from the intense monitoring period. Also we have performed data-resampling using jackknifing \\citep{shao95} by selecting randomly sub-samples from the data set and binning them in the flux\/photon index plane. All the results are extremely consistent with each other. Finally, in order to ensure that the overall anti-correlation trend is not induced by any sort of statistical dependence between the flux and $\\Gamma$ \\citep[e.g.][]{vaughan01}, we measured the eccentricity and the orientation of the $\\chi^2$ contour plots of the fits, used to derived the uncertainties of the fluxes and slopes. The 68 per cent ellipsoids have an average eccentricity of $0.91\\pm0.03$ and they are tilted on average by an angle of $(79\\pm2)\\degr$ counter-clockwise from the horizontal (flux) axis favouring, if at all, trends moving the opposite direction i.e.\\ a positive correlation between flux and $\\Gamma$. We therefore conclude that for NGC\\,7213, contrary to what is observed in other luminous Seyfert galaxies, the X-ray photon index anti-correlates with the X-ray flux. \n\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{figures\/linRegress2dCI.eps}\n\\caption{The confidence contours for the best fitting model of Fig.~\\ref{fig:rxte_hard_bright} taking into account the uncertainties in both coordinates. The black thin lines correspond to the 99, 95 and 68.3 confidence contours and represent the ellipsoids having from the minimum a $\\Delta\\chi^2$ of 6.63, 3.84 and 1, respectively, for 1 d.o.f. The grey thick line corresponds to the the 68.3 per cent joint confidence contour for the slope and the intercept having from the minimum a $\\Delta\\chi^2$ of 2.30, for 2 d.o.f. The horizontal and vertical dashed grey lines correspond to the tangents of the 68 per cent confidence region yielding the 68 per cent uncertainty ranges of the slope and intercept respectively.}\n\\label{fig:rxte_linRegrConf}\n\\end{figure}\n\n\n\\subsection{Hardness Ratio Analysis}\n\\label{ssect:hr}\nA completely model independent way to check for the validity of the above-mentioned anti-correlation behaviour, is by estimating the hardness ratio from the X-ray light curves. After binning the light curves in bins of 50 consecutive observations, we estimate the hardness ratio (5--10 keV)$\/$(2--4 keV) versus the overall count-rate in 2--10 keV. In Fig.~\\ref{fig:rxte_hardnessRatio} we plot the corresponding estimates, showing a clear increasing trend which implies that the X-ray behaviour of the source becomes `harder' when the source gets brighter.\\par\nWe can now fit to the hardness ratio data a linear model, $(y=\\alpha x+\\beta)$, taking into account the errors in both coordinates (as in Section \\ref{ssect:rxte_lc}). The best fit model has a slope of $\\alpha=0.11\\pm0.01$ and an intercept $\\beta=1.81\\pm0.02$ yielding a $\\chi^2$ of 4.31 for 15 d.o.f. having a null hypothesis probability of $3.4\\times10^{-3}$. Since the 99 per cent confidence interval for the slope is (0.07,0.14) the null hypothesis can be rejected at 1 per cent significance level. Finally, the value of ${\\itl t}$-statistic that we get from the data is 10.1\\footnote{The value of ${\\itl t}$-statistic is ${\\itl t}_{15,0.005}=2.95$}, corresponding to a probability of getting such a value from chance alone equal to $4.6\\times10^{-8}$. Therefore, we can robustly conclude that the best-fit slope is significantly different from zero.\n\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{figures\/rxte_hardnessRatio.eps}\n\\caption{Hardness ratio plot of the count-rate in (5--10 keV)$\/$(2--4 keV) versus 2--10 keV. The bin size corresponds to 50 consecutive observations. The solid line indicates the best-fit linear model to the data taking into account the uncertainties in both coordinates. The positive slope implies a `harder when brighter' X-ray spectral behaviour.}\n\\label{fig:rxte_hardnessRatio}\n\\end{figure}\n\n\n\\subsection{The nuclear broad-band Spectral Energy Distribution}\n\\label{ssect:SED}\nThe average spectral energy distribution of NGC\\,7213\\ (in $\\nu L_{\\nu}$ representation) is shown in Fig.~\\ref{fig:SEDngc7213} (both panels) with black symbols corresponding to flux estimates (listed in Table \\ref{tab:fluxEntries}) derived in this work and and grey symbols to archival data. The error bars correspond to the standard deviation around the mean flux, at a given frequency, whenever multiple observations from different epochs are available. In this way, the error bars are indicative of the actual variability of NGC\\,7213 (the actual uncertainty of single pointing observations is much smaller than the symbol size). The average flux from the ensemble of \\textit{RXTE} observations in the 2--10 keV energy range is shown by symbol 1, and corresponds to a power law having normalization of $(6.51\\pm3.73)\\times 10^{-3}$ photons s$^{-1}$ cm$^{-2}$  keV$^{-1}$ (at 1 keV) and a photon index of $1.87\\pm0.23$. The bowtie 2 indicates the 0.1--10 keV spectrum from the \\textit{XMM-Newton} observation (Emmanoulopoulos et al.\\,in prep.) which is broadly consistent with a power law with a normalization $2.99^{+0.06}_{-0.03}\\times10^{-3}$ photons s$^{-1}$ cm$^{-2}$ keV$^{-1}$ (at 1 keV) and a photon index $1.88^{+0.04}_{-0.02}$. The bowtie 3 indicates the average flux estimate registered by the \\textit{Swift}-BAT instrument, between 15--150 keV, corresponding to a power law with normalization\\footnote{The flux value is consistent with the one derived from the maps of the 4\\textsuperscript{th} \\textit{IBIS\/ISGRI} soft $\\gamma$-ray survey catalogue \\citep{bird10}.} $5.4^{+3.1}_{-2.0}\\times 10^{-3}$ photons s$^{-1}$ cm$^{-2}$ keV$^{-1}$ (at 1 keV) and a photon index of 1.82$^{+0.13}_{-0.12}$.  Finally, the 0.1--100 GeV the \\textit{Fermi}-LAT upper limit, assuming a photon index of 1.75, is $3\\times10^{-9}$ photons s$^{-1}$ cm$^{-2}$, is indicated by the arrow at 0.25 GeV (mean energy of the {Fermi}-LAT energy band).\\par\nThe mean SED of NGC\\,7213 is shown in Fig.~\\ref{fig:SEDngc7213}. This is true even for the {\\it RXTE} and {\\it Swift-}BAT data, since at these flux levels the contribution of any host galaxy emission should be minimal, especially in the {\\it Swift-}BAT band, as well as for the optical data, where we have carefully discard the host galaxy contribution from our measurements. Also note that, the optical, {\\it RXTE} and {\\it Swift}-Bat data plotted on this figure are indicative of the source flux averaged over many years, which are largely overlapping. On the other hand, the beam size of the radio observations is rather large, due the short exposure times (Section \\ref{ssect:sed_data}). As a result, the observed radio fluxes we have used may be contaminated by radio emission from a substantial part of the host galaxy itself. However, our radio flux estimates, at least for the 4.8 and 8.6 GHz bands, are identical to the mean flux values reported by \\citet{bell11}, who used a beam size of $0.5\\arcsec$ for a 12 h integration time. In addition, the observed variability at all radio bands indicate that most of the emission should originate from the nucleus itself. Detailed X-ray studies \\citep[e.g. Emmanoulopoulos et al.\\,in prep.,][]{lobban10} indicate that the intrinsic absorption towards the nuclear source in NGC\\,7213 is minimal in addition to the fact that the galaxy of NGC\\,7213 is face-on.\\par\nConsequently, the SED in Fig.~\\ref{fig:SEDngc7213} should be representative of the mean-intrinsic nuclear SED of NGC\\,7213. \nThe blue solid and dashed-line in the top panel of Fig.~\\ref{fig:SEDngc7213} indicates the average SED of radio-quiet and radio-loud quasars \\citep{elvis94}, respectively, and the blue open-circles in the bottom panel of the same figure indicate the average SED of LINERs \\citep{eracleous10}. Their relative position is set by minimizing the distance of their logarithmic ordinates from those of NGC\\,7213's, at the same frequencies, weighted by their squared errors. Since the distance minimization procedure is done in the logarithmic plane, in this way we estimate effectively an optimum normalization for each average-SED. Our results suggest that it is the LINERs mean SED which fits best, within the NGC\\,7213's SED. We therefore conclude that the nuclear continuum emission of the source is indeed representative of the mean SED of nearby LINER's.\\par \nDespite the fact that the (archival) near-IR flux measurements of NGC\\,7213 (as shown in the bottom panel of Fig.~\\ref{fig:SEDngc7213}) are representative of emission from a central region which is less than $0.35\\arcsec$ in size, it is not clear whether they correspond to emission form the nuclear source itself. For example, if we indeed observe the active nucleus directly, without any significant absorption, the detected near-IR emission cannot be due to optical\/UV nuclear emission being absorbed by the putative dusty, obscuring torus in this galaxy and re-emitted in the near-IR, because the average optical luminosity is significantly lower than the observed near-IR luminosity. On the other hand, NGC\\,7213 shows evidence of a  starburst driven wind \\citep{bianchi08} and hosts a star-forming ring few kpc away from the nucleus. A clumpy torus with toroidal shape \\citep{nenkova08}, having a small angular width parameter can then explain the observed infrared luminosity as being due to reprocessed radiation from star-forming regions, while the optical nuclear emission could escape unabsorbed through a torus-hole. Note that LINERs are thought to have relatively hot yet normal main-sequence O stars \\citep{shields92} able to heat the dust around them, something that strengthens even more the LINER interpretation of NGC\\,7213 SED.\\par\nThe SED plotted in Fig.~\\ref{fig:SEDngc7213} is based on more observations of the nuclear source flux, and is spread over a larger frequency range, than the SED presented by \\citet{starling05}. We can therefore use it to derive a more accurate estimate of the nuclear bolometric luminosity. After interpolating the SED linearly in logarithmic space, we integrate it between 1.344 GHz and $3.63\\times10^{19}$ Hz (i.e.\\ 150 keV) and we derive $L_{\\rm bol}=1.7\\times 10^{43}$ ergs s$^{-1}$, yielding an accretion rate of 0.14 per cent of Eddington limit.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{figures\/NGC7213_SED1.eps}\\\\[-1.73em]\n\\hspace*{0em}\\includegraphics[width=3.5in]{figures\/NGC7213_SED2.eps}\n\\caption{The SED of NGC\\,7213. For both panels, black symbols correspond to flux density estimates derived in this work and grey symbols correspond to archival data. The enumerated symbols, 1,2 and 3, correspond to the average RXTE spectrum (depicting the long-term variations), \\textit{XMM-Newton} and \\textit{Swift}-BAT spectrum, respectively. The grey arrow, corresponds to the \\textit{Fermi}-LAT upper-limit, in the 0.1-100 GeV energy range. (Top panel) The SED of NGC\\,7213 together with the average SEDs of radio-quiet and radio-loud quasars shown with the blue solid- and dashed-lines, respectively \\citep[taken from][]{elvis94}. (Bottom-panel) The SED of NGC\\,7213 together together with the average SEDs LINERs shown with blue open-circles \\citep[taken from][]{eracleous10}.}\n\\label{fig:SEDngc7213}\n\\end{figure}\n\n\\section{Discussion}\n\\label{sect:discussion}\nWe have analysed the long-term RXTE observations of NGC\\,7213 and we found that this low luminosity source exhibits a clear `harder when brighter' X-ray behaviour (Fig.~\\ref{fig:rxte_hard_bright}). This is the first time that such a spectral variability behaviour is reported for either a `low' or `high' luminosity AGN, and is contrary to what is observed in luminous Seyferts and quasars. We also constructed the average, nuclear SED of this source, using archival and proprietary data that we reduced, together with other measurements from the literature and we found that its shape closely resembles the mean SED of nearby LINERs. Finally, we provided a new measurement for the nuclear bolometric luminosity of NGC\\,7213: $L_{\\rm bol}=1.7 \\times 10^{43}$ ergs s $^{-1}$, yielding a rather low accretion rate of 0.0014 (i.e.\\ 0.14 per cent) of the Eddington limit.\\par \nThe average, broad band nuclear SED of NGC\\,7213 resembles that of LINERs (Fig.~\\ref{fig:SEDngc7213}), bottom panel). In the average SEDs of radio-loud and the radio-quiet AGN, the optical flux is quite higher than the X-ray flux. This is definitely not the case for this source, since the optical flux lies well below that of the X-rays. This is a very robust result, as both the optical and the X-ray data are based on long, monitoring observations, i.e.\\ they show the average behaviour of the source over many years. Furthermore, \\citet{wu83} noticed that the UV excess for NGC\\,7213 should be extremely week or absent, while both the recent {\\it XMM-Newton} observations (Emmanoulopoulos et al, in prep.), as well as the {\\it Suzaku} observations \\citep{lobban10} show no indication of soft X-ray excess, which is typical in most luminous AGN. All these results strongly suggest that the `big-blue bump' is missing in this source. This can be expected in a scenario in which the inner part of the geometrically thin and optically thick disc is missing in NGC\\,7213.\\par  \nOur results suggest that the accretion rate of the source is significantly smaller (by a factor of 10) than the `critical' rate at which accreting BHXRBs move from the `hard' to the `soft state'. Therefore, NGC\\,7213 could be the `hard state' analogue of BHXRBs, and in fact, its `harder when brighter' behaviour strongly supports this hypothesis. Although the global relationship between $\\Gamma$ and $\\xi$ in BHXRBs is well-established by comparing measurements from single-epoch observations \\citep[e.g.][]{wu08,younes11} the correlation of $\\Gamma$ with $\\xi$ on short time scales (within an observation, i.e.\\ down to minutes or even seconds) is less well-determined. However, \\citet{axelsson08} showed for the BHXRB Cyg\\,X-1 that the hardness-flux anti-correlation, seen in variations within an observation in brighter `hard' states, becomes a positive correlation in the faintest `hard' states.  This change in behaviour is consistent with a positive $\\Gamma$--$\\xi$ correlation at higher $\\xi$ changing to an anti-correlation at lower $\\xi$, i.e.\\ the variations within an observation follow the same trends as the global $\\Gamma$--$\\xi$ relationship.\\par\nTheoretically a considerable progress has been accomplished in this field with a better understanding of the complexity of the BHXRBs' `hard state' as well as LLAGN \\citep[see for a review,][]{narayan05}. A currently proposed model involves an accretion disc plus a hot accretion flow model, ADAF model, to explain the spectral dependence on accretion rate for BHXRBs, from quiescence up to the `soft state'. For intermediate accretion rates, as the accretion rate increases the Compton parameter in the hot accretion flow increases as well, producing a Comptonisation X-ray spectrum with a `harder' power law slope. This regime is identified with the `hard state', at mass accretion rates of 1--8 per cent of the Eddington rate, since the accretion process there is inefficient corresponding to lower fractions of L$_{\\rm Edd}$, consistent with our observations of NGC\\,7213. As the accretion rate increases further, the model predicts a different behaviour in which increasing luminosity corresponds to `softer' X-ray power law slopes. This regime is identified with the `soft state' and this is the what is normally observed in higher luminosity AGN.\\par\nFinally, another interesting feature of the source's SED is the fact that its X-ray emission above 20 keV is quite high, and implies a high-energy cut-off larger than 350 keV \\citep{lobban10}. NGC\\,7213 shows some week evidence of a radio jet structure at 8.4 GHz \\citep{blank05}. If such a structure is indeed there and it is aligned towards the observers direction it could produce relativistically amplified radio emission through synchrotron radiation, and enhanced X-ray emission through inverse Compton radiation of either the synchrotron photons and\/or optical photons from the star-burst environment of the host galaxy. This possibility could explain the fact that the radio emission of the source lies between that of radio-loud and radio-quiet AGN, similar to blazars. At the same time, the existence of a jet could also account for the `harder when brighter' behaviour of NGC\\,7213, something which is commonly observed in blazars \\citep[e.g.][]{krawczynski04,gliozzi06,zhang06} and can be explained in the framework of the synchrotron self Compton models. A jet model has also been proposed to explain the spectral evolution of the BHXRB XTE\\,J1550-564 as it moves from the fading state towards the X-ray `hard state' \\citep{russell10}.\\par\nTherefore, a jet pointing towards us, contributing significantly to the radio and X-ray emission of the source can not be ruled out, and in fact, for the case of ADAF models, it can be created through the Blandford-Znajek mechanism \\citep{armitage99}. In the `hard state' of BHXRBs the kinetic luminosity in the jet is believed to equal the radiation luminosity \\citep{fender10}. For the case of NGC\\,7213, assuming the existence of a jet, the previously derived accretion rate may therefore be lower by a factor of two, still remaining well below the typical transition between the `hard' and the `soft' state in BHXRBs. In the future, models combining a jet component with a geometrically thin disk and ADAF \\citep{yuan05} need to be tested for the case of NGC\\,7213.\n\n\\section*{Acknowledgments}\nDE and IMM acknowledge the Science and Technology Facilities Council (STFC) for support under grant ST\/G003084\/1. IP acknowledges support by the EU\nFP7-REGPOT 206469 grant. PA acknowledges support from Fondecyt grant number 11100449. This research has made use of NASA's Astrophysics Data System Bibliographic Services. Finally, we are grateful to the anonymous referee for the useful comments and suggestions that helped improved the quality of the manuscript.\n\n\\input{ngc7213Emmanoulopoulos.bbl}\n\n\\bsp\n\\label{lastpage}\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}